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\lfoot{Kenrick Bingham $\cdot$ \texttt{kenny ät hut.fi} $\cdot$
       Master's Thesis $\cdot$ Draft \tanaan}
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%%%%%%%%%%%%%%%% Front Matter %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\pagenumbering{roman}
\thispagestyle{empty}

TEKNILLINEN KORKEAKOULU \\
Teknillisen fysiikan ja matematiikan osasto \\
Teknillisen fysiikan koulutusohjelma
\luonnos{\hfill \fbox{DRAFT \tanaan}}


\vfill
\vfill

%\begin{tabbing}
%\hspace{3cm} \= Kenrick Bingham \\\\
%\> \textbf{Mathematical Background of Local X-Ray Tomography}
%\end{tabbing}
\begin{center}
Kenrick Bingham

\textbf{Mathematics of Local X-Ray Tomography}
\end{center}

\vfill
\vfill
\vfill
Diplomi-insinöörin tutkintoa varten tarkastettavaksi jätetty diplomityö

\begin{tabbing}
  Työn valvoja \quad \= professori Olavi Nevanlinna \\
  Työn ohjaaja       \> professori Erkki Somersalo
\end{tabbing}

Espoo 18.8.1998

{
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\newcommand{\tiivotsikko}[1]{\textbf{#1}}

\selectlanguage{finnish}
\newpage
\thispagestyle{plain}

\hspace*{-0.7cm}
\begin{tabular}{|lll|}
\multicolumn{1}{l}{}\vspace*{-1.4cm}\\
\multicolumn{3}{@{}l@{}}{\small {TEKNILLINEN KORKEAKOULU 
  \hfill DIPLOMITYÖN TIIVISTELMÄ}} 
\\
\hline 
\tiivotsikko{Tekijä:} & Kenrick Bingham & \\
\tiivotsikko{Työn nimi:} & 
\multicolumn{2}{l|}{Lokaalin röntgentomografian matematiikkaa} \\
\tiivotsikko{English title:} & 
\multicolumn{2}{l|}{Mathematics of Local X-Ray Tomography} \\
\tiivotsikko{Päivämäärä:} & 18.8.1998 
& \tiivotsikko{Sivumäärä:} \pageref{ViimeinenSivu}
\\[4pt]\hline
\tiivotsikko{Osasto:} & 
\multicolumn{2}{l|}{Teknillisen fysiikan ja matematiikan osasto} \\
\tiivotsikko{Professuuri:} & Mat-1 Matematiikka \hspace{1cm} &
\\[4pt]\hline
\tiivotsikko{Työn valvoja:} & 
\multicolumn{2}{l|}{Professori Olavi Nevanlinna} \\
\tiivotsikko{Työn ohjaaja:} & 
\multicolumn{2}{l|}{Professori Erkki Somersalo}
\\[4pt]\hline
\multicolumn{3}{|l|}{\parbox[t]{\tiivistelmanleveys}{
Työssä käydään läpi lokaalin röntgentomografian matemaattista teoriaa.
Tämä teoria osoittaa, milloin ja miten $n$-ulotteinen kappale ($n \ge 2$)
voidaan rekons\-truoida käyttämällä mittaustuloksia siitä, kuinka paljon
röntgensäteet vaimenevat kappaleen läpi kulkiessaan.
Matemaattisesti kyse on kompakti\-kantajaisen funktion $f: \Rn \to \R$
määrittämisestä sen integraaleista eri suoria pitkin.
\\[2ex]
Itse funktion $f$ rekontruktio on numeerisesti epästabiili toimitus,
mutta työssä johdetaan lähteen \cite{smith85:mathfound} mukaisesti kaavat, 
joiden avulla $e * f$ ja $\Lambda e * f$ voidaan rekons\-truoida stabiilisti.
Tässä $e * f$ ja $\Lambda e * f$ ovat eräiden oletusten vallitessa
funktioiden $f$ ja $\Lambda f = \Fi\big(|\xi| \, \hat{f}(\xi)\big)$ 
\hbox{approksimaatioita}, joissa pienimmät yksityiskohdat ovat sumentuneet. 
Rekons\-truktio\-kaavoja johdettaessa tarvittavaa
Calderón-Zygmund-teoriaa käydään myös läpi,
pitkälti teoksen \cite{neri71:singular} esitystapaa seuraten.
\\[2ex]
Funktio $\Lambda f$ antaa käyttökelpoista tietoa kappaleen
sisäisestä rakenteesta, 
sillä \hbox{Calderónin} pseudo\-diffe\-ren\-ti\-aali\-operaattori $\Lambda$
säilyttää epä\-jatkuvuus\-kohtien sijainnin. 
Tämä osoitetaan 
lähteestä \cite{ramm96:radon_and_lt} löytyvän hahmotelman mukaisesti 
näyttämällä, että funktioiden $f$ ja $\Lambda f$
aalto\-rintama\-joukot ovat samat. 
Todistuksessa ei tarvita pseudo\-differentiaali\-operaattoreiden 
teoriaa.
\\[2ex]
Eräs syy funktion $\Lambda f$ tarkastelemiseen on, että
sen likimääräinen rekons\-truktio onnistuu paikallisesti:
jos kiinnostuksen kohteena on vain osa kappaleesta,
rekonstruointia varten tarvitaan mittaukset ainoastaan niitä suoria pitkin, 
jotka kulkevat kiinnostavan alueen \hbox{läpi} tai aivan sen läheltä.
Funktioita $f$ ja $e * f$ ei voida rekonstruoida paikallisesti.
\\[2ex]
Kaksi stabiilisuustulosta, joita ei ole suoraan esitetty kirjallisuuslähteissä,
todistetaan myös. \hbox{Niiden} mukaan mittausvirheen $L^2$-normi rajoittaa
tasaisesti funktioiden $e * f$ ja $\Lambda e * f$ rekons\-truktioiden virheitä.
\\[2ex]
Lukijalta edellytetään reaalianalyysin, distribuutioteorian ja
Fourier-analyysin perus\-tietoja; 
näiden keskeisimpiä kohtia luetellaan liitteessä. 
Työssä esitetään todistukset kaikille tuloksille, joita
käytetään teorian johtamisessa,
lukuunottamatta edellä mainittuja esitietoja sekä 
kahta Rieszin muunnosten jatkuvuutta koskevaa lausetta,
joiden osalta viitataan lähteeseen \cite{ziemer89:weakly_diff}.
Näistä lauseista seuraa myös se, että lähteessä \cite{smith85:mathfound}
esiintyvä funktio\-avaruus $\Dxr$ onkin itse asiassa vain
neliö\-integroituvien funktioiden avaruus. 
Tästä seikasta ei löydy mainintoja aikaisemmassa kirjallisuudessa.
}}
\\ && \\[\pelkkaextrarowheighthyppy]
\hline
\tiivotsikko{Avainsanat:} & 
\multicolumn{2}{l|}{tomografia, tietokonetomografia, CT, lokaali, inversio-ongelmat,} 
\\[-4pt]
& \multicolumn{2}{l|}{singulaariset integraalit, Calderón-Zygmund-teoria} 
\\[4pt]\hline
\multicolumn{2}{|l}{\tiivotsikko{Ei lainata ennen:}} &
\tiivotsikko{Työn sijaintipaikka:} \hspace{2cm}
\\[4pt]\hline
\end{tabular}

\selectlanguage{english}
\newpage
\thispagestyle{plain}

\hspace*{-0.7cm}
\begin{tabular}{|lll|}
\multicolumn{1}{l}{}\vspace*{-1.4cm}\\
\multicolumn{3}{@{}l@{}}{\small {HELSINKI UNIVERSITY OF TECHNOLOGY
  \hfill ABSTRACT OF MASTER'S THESIS}} 
\\
\hline 
\tiivotsikko{Author:} & Kenrick Bingham & \\
\tiivotsikko{Title of thesis:} & 
\multicolumn{2}{l|}{Mathematics of Local X-Ray Tomography} \\
\tiivotsikko{Finnish title:} & 
\multicolumn{2}{l|}{Lokaalin röntgentomografian matematiikkaa} \\
\tiivotsikko{Date:} & $18^{\textup{th}}$ August, 1998 \hspace{1.3cm} 
& \tiivotsikko{Pages:} \pageref{ViimeinenSivu}
\\[4pt]\hline
\tiivotsikko{Department:} & 
\multicolumn{2}{l|}{Department of Engineering Physics and Mathematics} \\
\tiivotsikko{Chair:} & Mat-1 Mathematics & 
\\[4pt]\hline
\tiivotsikko{Supervisor:} & 
\multicolumn{2}{l|}{Professor Olavi Nevanlinna} \\
\tiivotsikko{Instructor:} & 
\multicolumn{2}{l|}{Professor Erkki Somersalo}
\\[4pt]\hline
\multicolumn{3}{|l|}{\parbox[t]{\tiivistelmanleveys}{
This thesis provides a self-contained account of the basic mathematical theory
of local x-ray tomography.
The theory shows when and how an $n$-dimensional object ($n \ge 2$) 
can be reconstructed,
using attenuation measurements of x-rays passing through the object.
Mathematically speaking, the problem is to determine 
a compactly supported function $f : \Rn \to \R$ from its line integrals.
\\[2ex]
The reconstruction of $f$ itself is a numerically instable operation,
but formulae are derived for the stable reconstruction of 
$e * f$ and $\Lambda e * f$, following \cite{smith85:mathfound}. 
Here $e * f$ and $\Lambda e * f$ are, under certain assumptions,
blurred approximations of $f$ and
$\Lambda f = \Fi \big( |\xi| \, \hat{f}(\xi)\big)$, respectively.
The Calderón-Zygmund theory 
needed for deriving the reconstruction formulae
is also presented, largely according to \cite{neri71:singular}.
\\[2ex]
The function $\Lambda f$ provides meaningful information about
the internal structure of the object,
since the Calderón pseudodifferential operator $\Lambda$
preserves the locations of discontinuities.
This is shown by proving that the wave front sets of 
$f$ and $\Lambda f$ are the same. 
The elementary proof of this fact, which does not use the theory of
pseudodifferential operators, is presented as outlined in
\cite{ramm96:radon_and_lt}.
\\[2ex]
One reason for considering $\Lambda f$ is that
its approximate reconstruction can be done locally.
This is to say that if the region of interest is only part of the object
examined, measurements are needed only along lines 
through the region of interest, or very close to it.
The functions $f$ and $e * f$ cannot be reconstructed locally.
\\[2ex]
Two stability results not directly given in the references are proved.
They state that the errors in the reconstructed $e * f$ and $\Lambda e * f$ 
are uniformly bounded by the $L^2$ norm of the error in the measurements.
\\[2ex]
The presentation is self-contained in the sense that
only some basic knowledge of real analysis, distribution theory and
Fourier analysis is required; 
some of the most central results of this background theory
are listed in an appendix for reference. 
Apart from these prerequisites, proofs of all results used are given, 
except for two theorems concerning 
the continuity of the Riesz transform,
for whom the reader is referred to \cite{ziemer89:weakly_diff}.
These theorems also imply that the function space $\Dxr$, 
considered in \cite{smith85:mathfound},
is just the space of square-integrable functions.
Earlier statements of this fact were not found in literature.
}}
\\ && \\[\pelkkaextrarowheighthyppy]
\hline
\tiivotsikko{Keywords:} & 
\multicolumn{2}{l|}{tomography, computed tomography, CT, local, inverse problems,} 
\\[-4pt]
& \multicolumn{2}{l|}{singular integrals, Calderón-Zygmund theory} 
\\[4pt]\hline
\multicolumn{2}{|l}{\tiivotsikko{Not borrowable till:}} &
\tiivotsikko{Library code:} 
\\[4pt]\hline
\end{tabular}
}

\newpage

\chapter*{Preface}

The process of writing this thesis has been rewarding and enriching.
I wish to express my gratitude to Professor Erkki Somersalo
for his expert guidance,
and to Professor Olavi Nevanlinna for providing me with 
the time needed and an interesting and stimulating environment.

Numerous colleagues have made this task a pleasant and instructive one.
Lic.Tech.\ Samuli Siltanen deserves particular recognition 
for reviewing my manuscript 
and always being ready to engage in helpful discussions.
I am also much obliged to Ph.D.\ Juha Kinnunen for his kind advice
regarding the derivatives of the Riesz potential.

Last but not least, I thank my beloved wife Ella for
being an excellent companion on the way towards our Master's degrees.

\vspace{9ex}

\begin{tabbing}
\hspace{6cm} \=
Espoo, 18$^{\textup{th}}$ August, 1998
\\[7ex] \>
Kenrick Bingham
\end{tabbing}

\tableofcontents


%%%%%%%%%%%%%%%% Main part %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newpage
\pagenumbering{arabic}



\chapter{Introduction}
\label{chap:Introduction}


The need to find out the internal structure of an object 
without opening it physically arises in many situations. 
One such situation is medical imaging,
where the objective is to look inside the patient's body without
performing surgery.

A way to satisfy the need for noninvasive imaging was found in 1895 
when x-rays were discovered by Wilhelm Röntgen. 
They made it possible to see through the softer parts of different objects
and thus to produce a silhouette of, \emph{e.g.}, 
the skeleton of a living person.
The full three-dimensional structure remained, however, 
hidden from the outside observer for nearly a century.

The invention of \emph{computed tomography} (CT)\footnote{%
  The names \emph{computerized tomography},
  \emph{computer aided tomography} (CAT), 
  \emph{computerized axial tomography} (CAT) 
  and \emph{CAT-scan} are also used.}
in the beginning of the 1970's finally made it possible to reconstruct 
the three-dimensional structure of the object with a computer, 
using digitized x-ray pictures taken from different directions.
The intensity resolution with which the x-ray attenuation could be
analysed also increased substantially.
This made it possible to characterize different types of materials, 
or tissues, 
by the x-ray attenuation coefficient, 
which is also called the Hounsfield number
in honour of Sir Godfrey Hounsfield who developed the first CT device
in 1972.

Some of the mathematical foundations needed for the reconstruction 
have been known for a relatively long time. 
The reconstruction formula for a two-dimensional slice
was derived by Johann Radon as early as 1917 \cite{radon17}, 
but it was only with the development of computers 
that this reconstruction could be realized in practice.
%As is often the case, computational formulae and algorithms for
%the reconstruction were developed first, 
%and precise mathematical conditions under which the calculations
%were well defined were investigated somewhat later.

Computed tomography presents an \emph{inverse problem}.
If the three-dimensional structure of the object is known, 
calculating the two-dimensional projection in a particular direction
is a straightforward computation: one just needs to add up
the x-ray attenuation coefficients at all points along the straight line
along which the x-ray travels, to find out how much the ray will be
attenuated when it reaches the detector, photographic film 
or fluorescent plate. 
In the other direction, the task of reconstructing the three-dimensional
structure from the x-ray pictures, is a more complicated problem.

It is often the case that the region of interest is only a small
part of the object examined, for instance a particular organ
of a living person. 
In this case, it would be convenient to be able to do the reconstruction
\emph{locally}, that is,
only using x-rays passing through the region of interest or very close to it.
This would decrease both the x-ray dose and the amount of data that
must be processed (see Figure~\ref{fig:intro1}).
It turns out that the x-ray attenuation coefficient $f$ itself cannot
be reconstructed locally, but another function 
$\Lambda f = \Fi\big(|\xi| \, \Ff(\xi)\big)$ can, and that
for many practical purposes, $\Lambda f$ yields significant information
about the internal structure of the object.
\begin{figure}[htbp]
  \begin{center}
    \leavevmode
    \epsfig{file=pics/intro1.eps, width=10cm}
    \caption{Local tomography only uses data from x-rays passing through
      the region of interest or very close to it.}
    \label{fig:intro1}
  \end{center}
\end{figure}

This thesis attempts to give a self-contained account of
the basic mathematical theory showing when and how the
reconstruction can be done. The reader will be assumed to know
some real analysis, distribution theory and Fourier analysis,
some of the most central results of which will be revised in 
Appendix~\ref{app:MathematicalTools}.
Apart from these basic results, two theorems concerning the
continuity of the Riesz potential operators will also be stated
without proof in Section~\ref{sec:ContinuityOfRieszPotentials} of
Appendix~\ref{app:MathematicalTools}. 
The reason for omitting the proofs is that
they require a comparatively lengthy treatment
which does not directly support the rest of the work.
The proofs can be found in \cite{ziemer89:weakly_diff}.

The text is organised as follows.
Chapter~\ref{chap:notation} introduces the notation that will be used.
Chapter~\ref{chap:XRayImaging} derives the main results of this work
in a descriptive and non-rigorous way.
Chapter~\ref{chap:FourierTransformsOfSingularIntegrals}
calculates the Fourier transforms of the singular Riesz kernel,
presents some Calderón-Zygmund theory according to \cite{neri71:singular}
and applies it to calculate the Fourier transform 
of the Riesz transform, which is a singular convolution operator.

These Fourier transforms are used in Chapter~\ref{chap:ReconstructionFormulae}
for deriving the reconstruction formulae, 
following the treatment in \cite{smith85:mathfound}.
As the exact reconstruction of the x-ray attenuation coefficient $f$
turns out to be an instable operation, 
an approximate reconstruction formula for $e * f$, where 
$e$ is a blurring kernel, is derived too, as well as
a local reconstruction formula for $\Lambda e * f$, 
which can be seen as an approximation of $\Lambda f$.
Stability results for these approximate reconstructions
are derived at the end of the chapter
from the reconstruction formulae given in \cite{smith85:mathfound}.
It is also noted that the space $\Dxr$, 
appearing in the assumptions of many of the theorems,
is actually just the space of square-integrable functions,
as follows from the two theorems in \cite{ziemer89:weakly_diff}
mentioned above.

Chapter~\ref{chap:EquivalenceOfWaveFronts} shows that
$f$ and $\Lambda f$ have the same discontinuities, 
following the treatment in \cite{ramm96:radon_and_lt}.
Chapter~\ref{chap:conclusions} 
lists some further results describing the properties of $\Lambda f$,
presents a few extensions to the theory and 
proposes possible subjects for further study.

Computed tomography of course involves many other, 
both theoretical and practical issues, which we shall not go into 
but rather leave as possible subjects for further study.
Such issues relate, for instance, to
the numerical implementation of the reconstruction,
using some algorithm like the algebraic reconstruction technique (ART)
or the filtered backprojection algorithm,
and statistical questions relating to the measurement process.
The choice of the blurring kernel $e$ involves important numerical
considerations, too. Convolution with $e$ can usually be seen
as a low-pass filtering process on the image, 
and other digital image processing operations may be expedient 
for making the reconstructed image as informative as possible.
\cite{cho-jones-singh93,natterer86}

We shall also not consider other types of tomography, 
like magnetic resonance imaging (MRI, NMR),
electric impedance tomography (EIT), ultrasound imaging, 
magnetoencephalography (MEG) or
nuclear emission tomography, all of which are lively research fields.
On the other hand our treatment covers imaging using
any beams of rays or particles that travel along
a straight line and whose attenuation is measured behind the object,
because the mathematical model describing them is identical to that
of x-ray imaging. 
Such beams include gamma rays, which are in wide use in medical imaging, 
and electron beams.

Electron beams are used in electron microscopy, 
where the object examined is normally a planar specimen 
through which the beam must pass more or less transversally.
Therefore the attenuation of the beam can be measured only
in a limited range of angles,
which presents an incomplete data problem.
Other types of incomplete data problems are encountered when
an opaque implant prevents measurements through an area
within the region of interest, or when
only part of the object can be x-rayed, for example if the object is too big.
Questions concerning the well-posedness of incomplete data problems
and the uniqueness of their solutions
will also be left as topics for further study.

The ``colour spectrum'' --- 
the dependence of the x-ray attenuation coefficient on the
energy (wavelength) of x-rays used ---
will also be neglected, as well as all physical, technical and medical issues.





\chapter{Notational Conventions}
\label{chap:notation}


\subsection*{Sets of Numbers and Euclidean Spaces}

We shall use the notation
\begin{align*}
  \N &= \{0,1,2,\ldots\} \\
  \Z &= \{\ldots, -2, -1, 0, 1, 2, \ldots\} \\
  \Z_+ &= \{1,2,3,\ldots \}
\end{align*}
for the sets of natural numbers, integers and positive integers, 
respectively.
The real field will be denoted by $\R$ and the complex field
by $\C = \{ x + i y \,|\, x, y \in \R \}$.
\label{def:N} \label{def:Z} \label{def:Zp} \label{def:R} \label{def:C}

\label{def:Rn}
The $n$-dimensional Euclidean space will be denoted by
\begin{equation}\nt
  \Rn = \Big\{ x = (x_1, x_2, \ldots, x_n) = \sum_{j=1}^n x_j e_j \,\big|\, 
  x_j \in \R \Big\}.
\ent\end{equation}
The natural basis is $\{ e_j \}_{j=1}^n$, where
$e_j = (0,0,\ldots,0,1,0,\ldots,0)$ with the 1 at the $j^{\text{th}}$ place. 
\label{def:UnitBasisVector}
Analogously, $\N^n$ will be the set of $n$-tuples of natural numbers,
\begin{equation}\nt
  \N^n = \Big\{ \alpha = (\alpha_1, \alpha_2, \ldots, \alpha_n)
  = \sum_{j=1}^n \alpha_j e_j \,\big|\, \alpha_j \in \N \Big\}.
\ent\end{equation}

Inner products in $\Rn$ will be denoted by 
\begin{equation}\nt
  x \cdot y = \sum_{j=1}^n x_j y_j
\ent\end{equation}
and norms by
\begin{equation}\nt
  |x| = \sqrt{x \cdot x}.
\ent\end{equation}

The letters $r$, $\theta$, $\varphi$ and $\omega$ will often be used
without separate mention to denote the polar and spherical coordinates 
introduced in Section \sectref{sec:PolarCoordinates} 
of Appendix~\ref{app:MathematicalTools}.

All integrals appearing will be Lebesgue integrals
and all functions will be Lebesgue measurable.
If the domain of integration is omitted, it is assumed to be the entire space.
The Lebesgue measure of the obvious dimension will be denoted by $m$;
in the zero-dimensional case, we shall use the counting measure. 
By saying that a condition $P(x)$ holds ``\emph{for almost all $x$}''
or ``\emph{almost everywhere}'' (abbreviated ``\emph{a.e.}''), we mean
that there exists a set $N$ such that $m(N)=0$ and that 
$P(x)$ holds for all $x \in \Rn \setminus N$.

For all $n \in \Z_+$, we shall denote the unit ball in $\R^n$
by $B^n$ and its boundary, the unit sphere in $\R^n$, by $S^{n-1}$:
\label{def:UnitBall} \label{def:UnitSphere}
\begin{equation}\nt
  B^n = \{x \in \R^n \,|\, |x| < 1\}, \qquad\quad
  S^{n-1} = \{x \in \R^n \,|\, |x|=1 \} = \partial B^n
\ent\end{equation}
(the notation $\partial A$ is used for the boundary of the set $A$).
\label{def:boundary}
Their $n$-dimensional and $n-1$-dimensional Lebesgue measures,
respectively,
are calculated in Lemma~\ref{lemma:MeasureUnitSphereAndBall}:
\begin{equation}
  \label{MeasureUnitSphereAndBall}
  |B^n| := m(B^n) = \frac{2 \pi^{n/2}}{n \,\Gamma\left(\frac{n}{2}\right)} 
  \quad \text{and} \quad
  |S^{n-1}| := m(S^{n-1}) 
    = \frac{2\pi^{n/2}}{\Gamma\left(\frac{n}{2}\right)} = n\,|B^n|.
\end{equation}
The unit hemisphere closest to $y \in \Rn \setminus \{0\}$
will be denoted by
\label{def:hemisphere}
\begin{equation}
  S^{n-1}_+(y) = \{ x \in S^{n-1} \,|\, x \cdot y > 0\}.
\end{equation}
More generally, the open ball and sphere 
with radius $R$ and centre at $x_0$ will be denoted, respectively, by
\label{def:sphere}
\begin{align}
  B^n(x_0,R) &= \{ x \in \R^n \,|\, |x-x_0| < R \} \\
\intertext{and}
  S^{n-1}(x_0, R) &= \{ x \in \R^n \,|\, |x-x_0| = R \}.
\end{align}
\label{def:BallRadiusRCentreX0}%
The dimension $n$ will in most cases be omitted from the notation
when it is obvious.
For $k \le n$, we shall often think of $\R^k$ as being embedded into $\R^n$.

The distance between a point $x \in \Rn$ and a set $B \subset \Rn$
and the distance between two sets $A,\, B \subset \Rn$ are denoted by 
\begin{equation}
  \dist(x, B) = \inf_{y \in B} |x - y|
  \qquad \text{and} \qquad
  \dist(A, B) = \inf_{x \in A} \dist(x, B),
\end{equation}
respectively. \label{def:dist}
The characteristic function of a subset $X \in \Rn$ is
the function $\chi_X : \Rn \to \{0,1\}$ defined by
\label{def:CharacteristicFunction}
\begin{equation}\nt
  \chi_X(x) = 
  \begin{cases}
    1, & x \in X \\
    0, & x \not\in X.
  \end{cases}
\ent\end{equation}


The symbol $:=$ will be used in definitions 
in the middle of the text, for emphasizing that we are making a definition
and not asserting an equality.
For example $a := b$ will mean that $a$ is defined to be equal to $b$.
Analogously, $a =: b$ will mean that $b$ is defined to be equal to $a$.
\label{def:def}


\pagebreak
\subsection*{Multi-indices}

Multi-indices
$\alpha = (\alpha_1, \alpha_2, \ldots, \alpha_n) \in \N^n$
will often be used to write monomials of $x \in \R^n$:
\begin{equation}\nt
  x^\alpha = x_1^{\alpha_1} x_2^{\alpha_2} \cdots x_n^{\alpha_n}
           = \prod_{j=1}^n x_j^{\alpha_j}.
\ent\end{equation}
The total degree of a multi-index is 
\begin{equation}\nt
  |\alpha| = \sum_{j=1}^n |\alpha_j|.
\ent\end{equation}
An analogous notation will be used for classical partial derivatives:
\begin{equation}\nt
  \partial^\alpha f 
  = \dodo{^{\alpha_1}}{x_1^{\alpha_1}} \dodo{^{\alpha_2}}{x_2^{\alpha_2}}
    \cdots  \dodo{^{\alpha_n}}{x_n^{\alpha_n}} \,f
  = \dodo{^{|\alpha|} f}{x_1^{\alpha_1} \partial x_2^{\alpha_2} \cdots
             \partial x_n^{\alpha_n}}.
\ent\end{equation}
Hence, the total degree coincides here with the order
of the partial derivative. \label{def:PartialDerivative}
In first-order partial derivatives, $\partial_{e_j}$ is abbreviated
as $\partial_j$.
For distribution derivatives (weak derivatives), the letter
$D$ will be used instead of $\partial$.

The Leibniz rule for multi-indices takes the form
\begin{equation}
  \partial^\alpha (uv)
  = \sum_{\beta \le \alpha}
    c_{\alpha\beta}\, \partial^\beta u\, \partial^{\alpha - \beta} v,
\end{equation}
where $c_{\alpha\beta} \in \N$ are constants and
$\beta \le \alpha$ means the partial order 
$\beta_j \le \alpha_j$ for all $j \in \{1, \ldots, n\}$.


\subsection*{Function Spaces and Distributions}

The most important mathematical tools that will be used are the basic
results of real analysis, the theory of distributions and Fourier analysis. 
Some of the most central ones are presented for reference in
Appendix~\ref{app:MathematicalTools}. 
The theory can be found in references
\cite{friedlander82:distr,hewitt-stromberg:real_and_abstract,hormander,
  rauch91:pde,rudin1,rudin2}.

We shall be dealing with the following function spaces, all of which
have a vector space structure:
\begin{itemize}
  \item $C^k(X)
%         = \{f:X \to \C \,|\, 
%         \forall \, x \in X,\, \alpha\in \N^n : \: 
%         |\alpha| \le k \!\!\imp\!\! \exists\, \partial^{\alpha} f \}
        $,
        the space of $k$ times continuously differentiable functions,
        $k \in \N \cup \{\infty\}$
%  \item $C(X) = C^0(X)$, the space of continuous functions
  \item $C_0^k(X) = \{f \in C^k(X) \,|\, \supp f \text{\,compact}\}$,
        the space of compactly supported $C^k$ functions
  \item $L^p(X) = \{f:X \to \C \,|\, \nl{f}{p} < \infty \}$, 
        with norm $\nl{f}{p} = \left(\int_X |f(x)|^p \,dx\right)^{1/p}$,
        $1 \le p < \infty$ (see Section~\sectref{sec:Lp})
  \item $L^1(X)$, the space of integrable functions
  \item $L^\infty(X) = \{f:X \to \C \,|\, \nl{f}{\infty} < \infty\}$,
        the space of essentially bounded functions,
        with norm 
        $\nl{f}{\infty} = \inf \{M \in \R \,|\, f(x) \le M \text{\,a.e.}\}$
  \item $L^p_0(X) = \{f \in L^p(X) \,|\, \supp f \text{ compact} \}$,
        the space of compactly supported $L^p$ functions
  \item $L^p_{\textup{loc}}(X) 
        = \{f:X \to \C \,|\, K\subset X \text{ compact} \!\!\imp\!\!
                                  \int_K |f(x)|^p\,dx < \infty \}$,
        $1 \le p < \infty$
  \item $\Lloc(X)$, 
        the space of locally integrable functions
  \item $\DD(X),\, \DD'(X),\, \Ss(X),\, \Ss'(X)$, 
        the spaces of compactly supported test
        functions, distributions, rapidly decreasing functions and
        tempered distributions (see Section \sectref{sec:Distributions})
\nopagebreak
  \item $H^s(\Rn) = \{ f \in \Ss' \,|\, \norm[H^s]{f} < \infty \}$,
        with norm $\norm[H^s]{f} = \nl{(1+|\xi|^2)^{s/2}\, \Ff(\xi)}{2}$,
        Sobolev space of order $s \in \R$
%        ($\Ff$ is the Fourier transform of $f$, 
%        see Section~\sectref{sec:FourierTransform})
        (see Section~\sectref{sec:Hs}).
\end{itemize}
\label{def:Ck} \label{def:Ck0} \label{def:Lp} \label{def:Lploc} \label{def:Hs}
If the set $X \subset \Rn$ is omitted, 
it is assumed to be the whole $\Rn$.

The inner product of $f \in L^2(X)$ and $g \in L^2(X)$
will be denoted by
\begin{equation}
  \ip{f}{g} = \int_X f(x) \, \conj{g(x)} \,dx.
\end{equation}
The notation $\du{f}{\phi}$ will be used for the integral
$\int f(x)\phi(x) \,dx$, and more generally for the
distribution duality of the distribution $f$ and the test function $\phi$.
Observe that the complex conjugate of the second argument is taken
in $\ip{f}{g}$ but not in $\du{f}{g}$.

Of the various variants of the definition of the Fourier transform, 
we choose
\begin{equation}\nt
  \F f (\xi) = \Ff(\xi) = \Fm \int f(x)\, e^{-i x \cdot \xi} \,dx
\ent\end{equation}
for functions  $f \in \Ss$. For the definition for more general functions 
and further properties of the Fourier transform, see Section
\sectref{sec:FourierTransform}.

Various constants whose exact values are not of interest, 
will be denoted by $C_j$, $j \in \N$.

For other notation, see the table of notation 
in Appendix \sectref{app:notation}.



\chapter{X-Ray Imaging}
\label{chap:XRayImaging}


\section{Divergent Beam Radiography}

Consider an object in $n$-dimensional space, the inside structure of which
is to be investigated using x-ray imaging. 
(For the rest of this chapter, $n \ge 2$ can be considered fixed; 
in practice the value of $n$ is usually, of course, 2 or 3.)
Call $f(x)$ the x-ray attenuation coefficient at point $x \in \Rn$;
we expect $f$ to give information about the object
since the attenuation coefficient depends 
on the material through which the ray passes.

Suppose that the object is contained in a ball of radius $R$
with the centre at the origin, and that the x-ray attenuation coefficient
$f$ is zero outside the object:
\begin{equation}\nt
  \label{suppf}
  \supp f \subset B(0,R).
\ent\end{equation}
If the object is x-rayed in a direction $\theta \in \Sm$ 
from a point $a \in A := \Sm(0,R)$, 
the attenuation of the x-ray intensity $I$ at each point 
$a + t \theta$, $t \ge 0$, is
\begin{equation}
  \label{XRayAttenuationDiffEq}
  -dI = f(a + t \theta) \, I \, dt.
\end{equation}
By solving this differential equation, we see that the intensity of the x-ray 
measured by a detector situated behind the object is
\begin{equation}\nt
  I_{\text{meas}} 
  = I_0\, \exp \left( -\int_0^\infty f(a + t \theta)\, dt \right).
\ent\end{equation}

Our aim will be to derive formulae for reconstructing $f(x)$ 
from the measurements $I_{\text{meas}}$, or equivalently from
\begin{equation}\nt
  \int_0^\infty f(a + t \theta)\, dt 
  = \ln \left( \frac{I_0}{I_{\text{meas}}} \right),
\ent\end{equation}
with different combinations of $a \in A$ and $\theta \in \Sm$.

%\begin{definition}
%\label{def:DivBeamRadiograph}
%  Let $f: \Rn \to [0, \infty)$ be a function with 
%  $\supp f \subset B(0,R)$ for some $R > 0$.
%  Then the function $\DD f: A \times \Sm \to [0, \infty]$,
%  \begin{equation}\nt
%    \label{SecondDaft}
%    \DD_a f(\theta) = \int_0^\infty f(a + t \theta)\, dt,
%  \ent\end{equation}
%  is called the \emph{divergent beam radiograph} of $f$.
%\end{definition}

\begin{definition}
  \label{def:DivBeamRadiograph}
  For a measurable function $f: \Rn \to \R$,
  the function $\DD f: A \times \Sm \to \R$,
  \begin{equation}\nt
    \label{SecondDaft}
    (a, \theta) \mapsto
    \DD_a f(\theta) := \int_0^\infty f(a + t \theta)\, dt,
  \ent\end{equation}
  is called its \emph{divergent beam radiograph}.
\end{definition}

In the case $n = 2$, which is naturally very important in practice, 
the divergent beam radiograph $\DD_a f(\theta)$
coincides with the Radon transform $\mathcal{R}f(\theta, a \cdot \theta)$,
which is by definition the integral of $f$ over a hyperplane:
\label{def:RadonTransform}
\begin{equation}
  \mathcal{R}f(\theta, s)
  = \int_{x \cdot \theta = s} f(x) \,dx.
\end{equation}
This transform is named after Johann Radon who already in 1917
derived an inversion formula for the two-dimensional Radon transform 
\cite{radon17}.
The theory of computed tomography is often presented using the
Radon transform instead of the divergent beam radiograph, 
and formulae are derived for reconstructing $f$ from $\mathcal{R}f$
\cite{natterer86,ramm96:radon_and_lt}.
We shall, however, do the generalisation into higher dimensions
using the divergent beam radiograph, 
sometimes also called the fan-beam transform in two dimensions and 
the cone-beam transform in three dimensions.

In practice, only a finite number of measurements are ever made.
The attenuation factors in different directions can be measured using
a spherical array of detectors on $A$. 
The x-ray source can then be turned together with the detectors, 
and the measurements repeated in each position. 
(See Figure~\ref{fig:DivBeamRadiography}.)
\label{DivBeamRadiographyDescription}

\kuvienpoisto{
\begin{figure}[htbp]
  \begin{center}
    \leavevmode
    \epsfig{file=pics/divergent.eps}
    \caption{Divergent beam radiography.}
    \label{fig:DivBeamRadiography}
  \end{center}
\end{figure}
}

The precise radiograph function $\DD f: A \times \Sm \to \R$ is thus
approximated by an interpolation of the measurements. 
Continuity results will be important in order to ensure
that this approximation and slight inaccuracies in the measurements 
do not cause large errors in the reconstructed x-ray attenuation coefficient.

The structure of a three-dimensional object can be computed
by working in $\R^3$, where the x-ray source must be moved 
about the whole surface of a sphere surrounding the object.
Another approach would be to ``slice'' the object up mathematically
in one direction, and to reconstruct the cross sections of the 
3-dimensional structure on each 2-dimensional slice, 
using measurements on a 1-dimensional sphere,
which is to say a circle, surrounding the slice in $\R^2$. 
The latter technique may be simpler to implement in practice,
with respect to both the mechanics of the measurement device
and the amount of data that must be processed in the computations.
However, both approaches have their own advantages.

In deriving the reconstruction formulae, let us first proceed formally,
and give rigorous proofs of the theorems in later chapters.
%Our treatment is largely based on \cite{smith85:mathfound},
%and we shall mostly use the same notation as it does.



\section{Exact Reconstruction Formula}
\label{sec:ExactReonctruction}

Integrating the measurements in all directions with the x-ray source
at a fixed point, we get, 
using polar coordinates $x = t \theta$, $dx = t^{n-1}\, dt\, d\theta$, 
\begin{equation}\nt
  \label{DaftIntegrated}
  \int_\Sm \DD_a f(\theta) \, d\theta = 
  \int_\Sm \int_0^\infty f(a + t \theta) \, dt \, d\theta =
  \int_\Rn f(a - x) \, |x|^{1-n} \, dx.
\ent\end{equation}
We shall write this in the form
\begin{equation}
\label{IntSmDaftEqualToR1ConvF}
  \int_\Sm \DD_a f(\theta) \, d\theta = \frac{1}{b_n} \, R_1 * f (a),
\end{equation}
using the Riesz potential:
\begin{definition}
  \label{def:RieszPotential} \label{def:RieszKernel}  
  For $n \ge 2$,
  the tempered distribution $R_1 \in \Ss'(\Rn)$, defined by the 
  locally integrable function
  \begin{equation}\nt
    \label{rieszpotential}
    R_1(x) = b_n\, |x|^{1-n}, 
    \qquad
    b_n = \frac{\Gamma(\frac{n-1}{2})}{2 \pi^{(n+1)/2}} 
        = \frac{1}{\pi\, |S^{n-2}|},
  \ent\end{equation}
  is called the \emph{Riesz kernel}.
  The function $R_1 * f$ is called the \emph{Riesz potential}
  of the measurable function $f$.
\end{definition}

Many texts, including \cite{natterer86}, \cite{ramm96:radon_and_lt},
\cite{stein70:sing} and \cite{ziemer89:weakly_diff}, 
use the notation $I_1 f$ or $I^1 f$ instead of $R_1 * f$.

Ignoring for a while that the measurements $\DD_a f(\theta)$ are only known
for $a \in A$, we perform the Fourier transform with respect to $a \in \Rn$ 
and get (the Fourier transform of $R_1$ will be calculated in Section 
\sectref{sec:FourierTransformOfRieszKernel})
\begin{equation}\nt
  \F \left(\int_\Sm \DD_a f(\theta) \,d\theta\right)
  = \frac{\Fp}{b_n} \, \F(R_1)\, \F f
  = \frac{1}{b_n} \, |\xi|^{-1} \Ff,
\ent\end{equation}
which leads to the reconstruction formula
\begin{equation}\nt
  f(x) = b_n\, \Fi \left(|\xi|\, 
           \F \left(\int_\Sm \DD_x f(\theta) d\theta \right)\right),
\ent\end{equation}
where $x$ is written instead of $a$.
This can be written more compactly as
\begin{equation}
  \label{DivRadiographReconstructionFormulaWithAllA}
  f(x)= b_n \, \Lambda \left( \int_\Sm \DD_x f(\theta) d\theta \right),
\end{equation}
where $\Lambda$ is the Calderón operator:
\begin{definition}
  \label{def:CalderonOperator}
  The operator 
  $\Lambda: 
%  \{f \in L^2(\Rn) \,|\, R_1*f \in L^2(\Rn) \} \to L^2(\Rn)$, 
%  \big\{ f \in \Ss'(\Rn) \,\big|\, 
%         |\xi| \, \Ff(\xi) \in L^2(\Rn) \big\} \to L^2(\Rn)$,
  \Ss'(\Rn) \to \Ss'(\Rn)$, $n \in \Z_+$,
  defined through its Fourier transform as
  \begin{equation}
    \label{def:Lambda}
    \F(\Lambda g)(\xi) = |\xi|\, \widehat{g}(\xi),
  \end{equation}
  is called the \emph{Calderón operator}.
\end{definition}

The formal calculation
\begin{equation}\nt
  \label{LambdaInvertsConvolutionByRieszKernelFormal}
  \F\big(\Lambda (R_1 * g)\big)(\xi) = |\xi|\, |\xi|^{-1} \widehat{g}(\xi) 
  = \widehat{g}(\xi)
\ent\end{equation}
above, when carried out properly in 
Chapter~\sectref{chap:ReconstructionFormulae},
yields the following result:
\begin{theorem}
  \label{thm:ExactReconstructionChap3}
  \repeated{Theorem~\ref{thm:ExactReconstruction}}
  If $n \ge 2$, $f \in \Dxr(\Rn)$ and $|\xi|^{-1} \Ff(\xi) \in \Lloc(\Rn)$,
  then for almost all $x$,
  \begin{equation}
    \label{ExactReconstructionFormula}
    \Lambda(R_1 * f)(x) = f(x).
  \end{equation}
\end{theorem}

The x-ray domain $\Dxr$ will be introduced in Definition~\sectref{def:Dxr}.

It will be convenient to know a more direct expression for the 
Calderón operator $\Lambda$, in addition to its Fourier transform.
The formal calculation
\begin{equation}
\begin{split}
  \Lambda f
  &
  = \Fi \Big( \sum_{j=1}^n - i \, \xi_j \, |\xi|^{-1} \, 
              i \, \xi_j \, \Ff(\xi) \Big)
  \\&
  = - \sum_{j=1}^n \Fm \, \Fi\big(i \, \xi_j \, |\xi|^{-1} \big)
                   * \Fi \big( i \, \xi_j \, \Ff(\xi) \big)
  \\&
  = - \sum_{j=1}^n \dodo{R_1}{x_j} * \dodo{f}{x_j}
%  \Lambda f 
%  &
%  = \Fi\big(|\xi|^2 \, |\xi|^{-1} \Ff(\xi)\big)
%  \\&
%  = - \laplace \, \Fi\big(|\xi|^{-1} \Ff(\xi)\big)
%  \\&
%  = - \laplace (R_1 * f)
%  \\&
%  = - (\laplace R_1) * f
%  \\&
%  = - \sum_{j = 1}^n \dodo{^2 R_1}{x_j^2} * f
%  \\&
%  = - \sum_{j = 1}^n \dodo{R_1}{x_j} * \dodo{f}{x_j}
\end{split}
\end{equation}
will be justified for $f \in L^2$ 
in Chapter~\ref{chap:FourierTransformsOfSingularIntegrals},
if the convolution is interpreted as a \emph{principal value} ($\pv$)
convolution, introduced in Definition~\sectref{def:pvConvolution}.
We then get the following theorem:
\begin{theorem}
  \label{thm:LambdaAspvConvolutionChap3}
  \repeated{Theorem~\ref{thm:LambdaAspvConvolution}}
  For all $n \in \Z_+$, the Calderón operator $\Lambda$ can be expressed as
  \begin{equation}
    \label{LambdaAsPVConvolutionChap3}
    \Lambda = - \sum_{j=1}^n \pv \dodo{R_1}{x_j} * D_j.
  \end{equation}
  It is a continuous operator from $H^1$ to $L^2$.
\end{theorem}


The problem of only knowing $\DD_a f(\theta)$ when $a \in A$ can now be
avoided by using the elementary relationship
\begin{equation}\nt
  \int_\Sm \int_{-\infty}^\infty g(x + t\theta) \, dt\, d\theta
  = 2 \int_\Sm \int_0^\infty g(x + t\theta) \, dt\, d\theta,
\ent\end{equation}
and making the substitution (see Figure~\ref{fig:subst2}) 
\begin{equation}
  \label{subst2}
  \theta = \frac{a - x}{|a - x|} \in \Sm,
  \qquad \qquad
  d\theta = \frac{|(x-a) \cdot a|}{R\, |x-a|^n} \, da
\end{equation}
\kuvienpoisto{%
\begin{figure}[htbp]
  \begin{center}
    \leavevmode
    \epsfig{file=pics/subst2.eps}
    \caption{The change of variable $\theta=\frac{a-x}{|a-x|}$, \,
      $d\theta = \frac{|\cos{\alpha}|}{|a-x|^{n-1}} \,da
      = \frac{|\frac{a-x}{|a-x|} \cdot \frac{a}{R}|}{|a-x|^{n-1}}
        \, da
      = \frac{|(x-a) \cdot a|}{R\, |x-a|^n} \, da$
      used for deriving \equ{DivRadiographReconstructionFormulaWithAOnA}.}
    \label{fig:subst2}
  \end{center}
\end{figure}}%
into \equ{DivRadiographReconstructionFormulaWithAllA}.
This yields the reconstruction formula
\begin{equation}
  \begin{split} \displaystyle
  \label{DivRadiographReconstructionFormulaWithAOnA}
  \displaystyle
  f(x) & = \tfrac{b_n}{2}\, \Lambda 
           \int_\Sm \int_{-\infty}^\infty f(x + t \theta) \,dt \,d\theta \\
       & = \tfrac{b_n}{2R}\, \Lambda 
           \int_A \int_{-\infty}^\infty f(x + t \,\tfrac{a-x}{|a-x|}) \,dt\, 
           \tfrac{|(x-a) \cdot a|}{|x-a|^n}\,da \\
       & = \tfrac{b_n}{2R}\, \Lambda \int_A \int_{-\infty}^\infty 
           f(a + (-|a-x|+t) \,\tfrac{a-x}{|a-x|}) \,dt\, 
           \tfrac{|(x-a) \cdot a|}{|x-a|^n}\,da \\
       & = \tfrac{b_n}{2R}\, \Lambda \int_A 
           \left(\DD_a f(\tfrac{a-x}{|a-x|}) 
             + \DD_a f(-\tfrac{a-x}{|a-x|}) \right) 
           \tfrac{|(x-a) \cdot a|}{|x-a|^n}\,da.
  \end{split}
\end{equation} % \cite[2.11]{smith85:mathfound}

However, there are problems with using this formula.
Since the operator $\Lambda$ involves differentiation and 
convolution with the strongly singular kernel $\dodo{R}{x_j}$,
the reconstruction is not stable with respect to errors in the measurements.
The continuity from $H^1$ to $L^2$ allows 
small but rapidly changing errors in the measurements to result
in large errors in the x-ray attenuation coefficient reconstructed.
An approximate reconstruction can be used instead with more success. 

The names of the commonly used \emph{filtered backprojection}
and \emph{backprojection filtering} algorithms 
\cite{natterer86,cho-jones-singh93}
come from the reconstruction formula \equ{ExactReconstructionFormula}:
The calculation of $R_1 * f$ by integrating the measurements
over the sphere, as in \equ{DivRadiographReconstructionFormulaWithAllA},
is the \emph{backprojection} operator. 
Multiplication on the Fourier transformed side, as by $|\xi|$ in $\Lambda$, 
is traditionally called \emph{filtering} in signal processing.
Some other filtering can be combined with it 
to alleviate the effects of noise. 
This gives an approximate reconstruction $e * f$, 
where $e$ is a blurring kernel.




\section{Parallel Beam Radiography}
\label{sec:ParallelBeamRadiography}

%When deriving an approximate reconstruction formula,
%we shall use the \emph{parallel beam radiograph}
%\label{def:ParBeamRadiograph}
%\begin{equation}\nt
%  \PP_\theta f(x) = \int_{-\infty}^\infty f(x + t\theta) \,dt
%\ent\end{equation}
%for
%\begin{equation}\nt
%  x \in \theta^\bot = \left\{ y \in \Rn \,|\, y \cdot \theta = 0 \right\},
%\ent\end{equation}
%which is related to the divergent beam radiograph through
%\begin{equation}
%  \label{PAndDRelation}
%  \PP_\theta f(E_\theta x) = \DD_x f(\theta) + \DD_x f(-\theta),
%\end{equation}
%where \label{def:OrthogonalProjection}
%\begin{equation}\nt
%  E_\theta (x) = x - (x \cdot \theta) \theta
%\ent\end{equation} 
%designates the orthogonal projection onto $\theta^\bot$, 
%$\theta \in \Sm \subset \Rn$.

When deriving an approximate reconstruction formula,
we shall use the \emph{parallel beam radiograph}:
\begin{definition}
  \label{def:ParBeamRadiograph}
  For a measurable function $f: \Rn \to \R$, the function 
  \begin{equation}
    \PP f: \{ (\theta, x) \in \Sm \times \Rn \,|\, x \in \theta^\bot \} \to \R,
  \end{equation}
  \begin{equation}\nt
    (\theta, x) \mapsto
    \PP_\theta f(x) := \int_{-\infty}^\infty f(x + t\theta) \,dt,
  \ent\end{equation}
  is called the \emph{parallel beam radiograph} of $f$.
\end{definition}

Here
\begin{equation}
  \label{def:OrthogonalComplement}
  \theta^\bot 
  := \{ \theta \}^\bot
  = \left\{ y \in \Rn \,|\, y \cdot \theta = 0 \right\}
\end{equation}
is the \emph{orthogonal complement} of $\theta$.


As is easily seen, the parallel beam radiograph is related 
to the divergent beam radiograph through
\begin{equation}
  \label{PAndDRelation}
  \PP_\theta f(E_\theta x) = \DD_x f(\theta) + \DD_x f(-\theta),
\end{equation}
where \label{def:OrthogonalProjection}
\begin{equation}\nt
  E_\theta (x) := x - (x \cdot \theta) \, \theta
\ent\end{equation} 
designates the orthogonal projection onto $\theta^\bot$, 
$\theta \in \Sm \subset \Rn$.

After using relation \equ{PAndDRelation} 
and passing the operator $\Lambda$ under the integral sign, 
\equ{DivRadiographReconstructionFormulaWithAllA} yields
the parallel beam reconstruction formula
\begin{equation}
  \label{ParBeamRadiographReconstructionFormula}
  f(x) = \frac{b_n}{2} \int_\Sm \Lambda \PP_\theta f(E_\theta x) \, d\theta.
\end{equation}

The parallel beam and divergent beam radiographs carry the same information
because $\DD_x f(\theta)$, $x \in A$, can be nonzero only when 
$\theta$ points towards the interior of the sphere $A$,
so that always at least one of the members of \equ{PAndDRelation} 
in known to vanish. 
Therefore $\PP f$ can be seen just as an alternative notation
that is more convenient in some cases. 
However, the parallel beam radiograph operator $\PP$,
sometimes also called the \emph{x-ray transform}, was perhaps more natural
in the historical setting where
information about the object was collected in a different way 
than that described above in Section \sectref{DivBeamRadiographyDescription}.

In parallel beam radiography, 
the object is exposed to a beam of parallel x-rays covering the whole object. 
(See Figure~\ref{fig:ParBeamRadiography}.)
The attenuation factor is measured at one point for each ray, 
and the measurements are repeated for all different directions 
$\theta \in \Sm$ of the beam. 

\kuvienpoisto{
\begin{figure}[htbp]
  \begin{center}
    \leavevmode
    \epsfig{file=pics/parallel.eps}
    \caption{Parallel beam radiography.}
    \label{fig:ParBeamRadiography}
  \end{center}
\end{figure}
}

Another possibility would be to use a single x-ray source 
and a single detector behind the object in direction $\theta$,
to scan over $\thb$ by moving the source and the detector together,
and to repeat this for all $\theta$. 
First generation CT scanners were based on this technique.

%Computed tomography first started with parallel beam radiography, but 
Since parallel x-rays are difficult to produce in practice,
and the latter method of scanning over each hyperplane $\thb$
can be quite time-consuming,
divergent beam radiography is nowadays more widely used
in computed tomography.



\section{Approximate Reconstruction Formula}
\label{sec:ApproxReconstruction}

The attenuation coefficient $f = \delta * f$ can be approximated 
by $e * f$, where $e$ is an approximate delta function, 
also called a \emph{blurring kernel} or \emph{point spread function},
for instance the one defined in \equ{def:ApproxDelta}.
For deriving a reconstruction formula for $e * f$, 
we shall use the following formal calculations:

The substitutions $z = y + t\theta \in \theta^\bot \oplus \R \theta = \Rn$, 
and $dz = dt\,dy$ and $\tau = t + s$, $d\tau = ds$ yield
\begin{equation}\nt
\begin{split}
  \left( \PP_\theta e * \PP_\theta f \right) (x) 
  &
  = 
  \int_{\theta^\bot} \infint e(y+t\theta)\,dt \infint f(x+s\theta-y)\,ds\,dy
  \\&
  = \int_\Rn e(z) \infint f\big(x-z+(t+s)\theta\big) \,ds \,dz
  \\&
  = \infint \int_\Rn e(z) f(x+\tau\theta-z) \,dz \,d\tau 
%  \\&
  = \PP_\theta (e * f) (x),
\end{split}
\ent\end{equation}
which in turn gives
\begin{equation}\nt
\begin{split}
\Lambda \big(\PP_\theta(e * f)\big) 
& = - \sum_{j=1}^n \pv \dodo{R_1}{x_j} * \dodo{}{x_j} 
        (\PP_\theta e * \PP_\theta f)
\\ &
  = - \sum_{j=1}^n \pv \dodo{R_1}{x_j} * 
        \Big(\dodo{\PP_\theta e}{x_j} * \PP_\theta f\Big)
\\ &
  = \Big( \!\! - \sum_{j=1}^n \pv \dodo{R_1}{x_j} * 
        \dodo{\PP_\theta e}{x_j} \Big) * \PP_\theta f
  = (\Lambda \PP_\theta e) * \PP_\theta f.
\end{split}
\ent\end{equation}
If we now replace $f$ by $e * f$ in Equation
\equ{ParBeamRadiographReconstructionFormula},
we get the approximate parallel beam reconstruction formula
\begin{equation}\nt
\label{ApproxParBeamReconstructionFormulaChap3}
\begin{split}
  e * f (x)
  = &\, \frac{b_n}{2} \int_\Sm \Lambda \PP_\theta (e * f) (E_\theta x)
     \,d\theta
\\  
 = &\, \frac{b_n}{2} \int_\Sm 
    (\Lambda \PP_\theta e) * \PP_\theta f (E_\theta x) \,d\theta
\\ 
 = &\, \frac{b_n}{2} \int_\Sm \int_{\theta^\bot} 
     \Lambda \PP_\theta e (E_\theta x - y) \, \PP_\theta f (y) \,dy \,d\theta.
\end{split}
\ent\end{equation}
We then perform the change of variable $y = E_\theta a$, 
$dy = \frac{|a \cdot \theta|}{R} da$. Here $y$ runs twice over
the set $B(0,R) \cap \theta^\bot$ (containing $\supp \PP_\theta f$)
as $a$ runs over the two hemispheres of $A$ 
(see Figure~\ref{fig:subst1}).
This gives the approximate divergent beam reconstruction formula
\begin{equation}\nt
\begin{split}
  \label{ApproxDivBeamReconstructionFormulaChap3}
  e * & f (x) 
  = \frac{b_n}{4R} \int_\Sm \int_A 
    \bigl(\Lambda \PP_\theta e (E_\theta (x - a)) \bigr) \, 
    \PP_\theta f (E_\theta a) \, |a \cdot \theta| \,da \,d\theta
\\ &
  = \frac{b_n}{4R} \int_A \int_\Sm
    \bigl( \Lambda \PP_\theta e (E_\theta (x - a)) \bigr) \,
    \bigl( \DD_a f (\theta) + \DD_a f (- \theta) \bigr) \,
    |a \cdot \theta| \,d\theta \,da.
\end{split}
\ent\end{equation}

\kuvienpoisto{
\begin{figure}[htbp]
  \begin{center}
    \leavevmode
    \epsfig{file=pics/subst1.eps}
    \caption{Substitution $y = E_\theta a$, 
      $dy = |\cos \alpha \,|\, da = \frac{|a \cdot \theta|}{|a|\,|\theta|}\, da
      = \frac{|a \cdot \theta|}{R}\,da$.}
    \label{fig:subst1}
  \end{center}
\end{figure}
}

These formulae will be derived in Chapter~\sectref{chap:ReconstructionFormulae}
under the following assumptions:

\begin{theorem}
  \label{thm:ApproxParBeamReconstructionChap3}
  \repeated{Corollary ~\ref{cor:ApproxReconstruction}}
  Let $n \ge 2$ and $e \in \Dxr(\Rn) \cap H^{1/2}(\Rn)$ be such that 
  $|\xi|^{-1} \Fe(\xi) \in \Lloc(\Rn)$. 
  If $f \in L^2_0(\Rn)$ and $\supp f \subset B(0,R)$, then
  \begin{equation}
  \begin{split}
    \label{ApproxReconstructionFormulaChap3}
    e * & f (x)
    = \frac{b_n}{2} \int_\Sm \int_{\theta^\bot} 
      \Lambda \PP_\theta e (E_\theta x - y) \, \PP_\theta f (y) \,dy \,d\theta
    \\&
    = \frac{b_n}{4R} \int_A \int_\Sm
      \Lambda \PP_\theta e \bigl(E_\theta (x - a)\bigr) \,
      \bigl( \DD_a f (\theta) + \DD_a f (- \theta) \bigr) \,
      |a \cdot \theta| \,d\theta \,da,
  \end{split}
  \end{equation}
  where $A = S^{n-1}(0,R)$.
\end{theorem}

In practice, it is of fundamental importance that 
this reconstruction is stable with respect to errors in the measurements:

\begin{theorem}
  \label{thm:ApproxReconstructionStabilityChap3}
  \repeated{Theorem~\ref{thm:ApproxReconstructionStability}}
  If $n \ge 2$ and $e \in \Dxr(\Rn) \cap H^{1/2}(\Rn)$ is such that 
  $|\xi|^{-1} \Fe(\xi) \in \Lloc(\Rn)$, 
  then there exists a constant $C > 0$ such that
  \begin{equation}
    |e * f(x)| 
    \le C \, \norm[L^2(\Sm(0, R) \times \Sm)]{\DD f}
  \end{equation}
  for all $f \in L^2_0(\Rn)$ with $\supp f \subset B(0,R)$ and all $x \in \Rn$.
\end{theorem}



\section{Reconstruction Formula for $\Lambda e * f$}
\label{sec:ReconstructionFormulaForLambdaEConvF}

For extracting information about the internal structure of the object,
$\Lambda e * f$ can also be reconstructed instead of $e * f$.
In that case, we have the following reconstruction formula,
which will also be derived in Chapter~\sectref{chap:ReconstructionFormulae}.

\begin{theorem}
  \label{thm:EConvLambdaFReconstructionChap3}
  \repeated{Theorem~\ref{thm:EConvLambdaFReconstruction}}
  Let $n \ge 2$ and 
  let $e \in H^{3/2}(\Rn)$ be such that $\Lambda e \in \Dxr(\Rn)$.
  If $f \in L^2_0(\Rn)$, then
  \begin{equation}
  \label{EConvLambdaFReconstructionFormulaChap3} 
  \begin{split}
    \Lambda(e * & f)(x) = (\Lambda e) * f(x)
\\
    = & - \frac{b_n}{2} \int_\Sm 
      \big(\laplace \PP_\theta e\big) * \big(\PP_\theta f\big)(E_\theta x) 
      \,d\theta
\\
    = & - \frac{b_n}{4R} \int_A \int_\Sm 
      \laplace \PP_\theta e(E_\theta (x - a))
      \left( \DD_a f(\theta) + \DD_a f(-\theta) \right) |a \cdot \theta|
      \,d\theta \, da
  \end{split}
  \end{equation}
  for all $x \in \Rn$.
  If, in addition, $e$ has compact support, then
  \begin{equation}
    \Lambda(e * f) = e * \Lambda f.
  \end{equation}
\end{theorem}

This reconstruction, too, is a stable operation:

\begin{theorem}
  \label{thm:EConvLambdaFReconstructionStabilityChap3}
  \repeated{Theorem~\ref{thm:EConvLambdaFReconstructionStability}}
  If $n \ge 2$ and 
  $e \in H^{3/2}(\Rn)$ is such that $\Lambda e \in \Dxr(\Rn)$,
  then there exists a constant $C > 0$ such that 
  \begin{equation}
    |\Lambda e * f(x)| 
    \le C \, \norm[L^2(\Sm(0, R) \times \Sm)]{\DD f}
  \end{equation}
  for all $f \in L^2_0(\Rn)$ with $\supp f \subset B(0,R)$ and all $x \in \Rn$.
\end{theorem}

Of course, $\Lambda e * f$ looks different from $e * f$, 
but in many practical applications it gives relevant information. 
Most importantly, 
$e * \Lambda f$ is nearly singular at points where $\Lambda f$ is singular.
These are precisely the same points at which $f$ is singular,
as will be shown in Chapter~\sectref{chap:EquivalenceOfWaveFronts}:
\begin{theorem}
  \label{thm:WaveFrontsEquivalentChap3}
  \repeated{Theorem~\ref{thm:WaveFrontsEquivalent}}
  If $n \ge 2$,
  $f \in L^2(\Rn)$ and $\Lambda f \in \Dxr(\Rn)$, 
  then $\WF{f} = \WF{\Lambda f}$.%
\end{theorem}

The \emph{wave front} set $\WF{f}$, describing the discontinuities of
the distribution $f$, will be introduced in Definition~\ref{def:WaveFront}.

This gives hope for the visibility of discontinuities of $f$ 
in $\Lambda e * f$. 
The fact that the discontinuities of these two functions
are located at the same points 
does not, of course, prevent the jumps of $\Lambda e * f$ 
from being much smaller and therefore undetectable in practice.
Fortunately, the situation is actually the opposite:
discontinuities of $f$ become emphasized in $\Lambda f$.

This fact and other further results will be described 
in the concluding Chapter~\ref{chap:FurtherPropertiesForLambdaF}.



\section{Local Tomography}
\label{sec:LocalTomography}

Reconstructing $e * \Lambda f (x)$ rather than $e * f(x)$ has the
advantage that it can be done using data only from beams passing 
very close to $x$, in which case the process is called \emph{local tomography}.
This is done by choosing the convolution kernel $e$ in such a way 
that for a fixed $x$, the support of the kernel 
$\laplace \PP_\theta e$
in the reconstruction formula~\equ{EConvLambdaFReconstructionFormulaChap3} 
is small.
Then $\DD_a f(\theta)$ is only needed for such $a$ and $\theta$ that
$E_\theta(x - a) \in \supp \laplace \PP_\theta e$.

An approximation of the x-ray attenuation coefficient $f$ itself 
cannot be reconstructed locally, because in the $e * f$ reconstruction 
formula~\equ{ApproxDivBeamReconstructionFormulaChap3}, the support of the
kernel $\Lambda \PP_\theta e$ may not be bounded even if 
the support of $e$ is.
This is a consequence of the fact that all pseudodifferential operators
\begin{equation}
  \label{PDOB}
  \mathcal{B} : f(x) \mapsto \Fi\big( p(x, \xi) \Ff(\xi) \big)
\end{equation}
that are \emph{local} in the sense that
\begin{equation}
  \supp \mathcal{B} f \subset \supp f
\end{equation}
are differential operators \cite{peetre59, peetre60}. 
Equivalently, for $\mathcal{B}$ to be local,
the function $p: \Rn \times \Rn \to \C$ in \equ{PDOB},
called the \emph{symbol} of $\mathcal{B}$, must be a polynomial in $\xi$,
possibly with functions of $x$ as coefficients.
In the case of $\Lambda$, 
\begin{equation}
  p(x,\xi) = |\xi| = \sqrt{\sum_{j=1}^n \xi_j^2}
\end{equation}
is clearly not a polynomial.

There are many practical benefits if the area of interest is 
only part of the object x-rayed, for example, the heart of a living person.

Because the x-ray beam can be kept narrower, the overall x-ray dose 
to which the object is exposed can be reduced, which is beneficial
since x-ray imaging is not \emph{completely} noninvasive;
adverse effects of x-ray exposure to both living creatures and 
inanimate objects are well known.

Fewer x-ray beams also imply a decrease in the amount of data 
that must be processed for reconstructing an image with a given resolution.
This speeds up the imaging process, 
which makes it possible to monitor the time evolution, 
for example, of a beating heart.
Alternatively, the resolution can be increased, 
since concentrating on a smaller region reduces the
technological restrictions arising from the design of the detectors
and the optical demagnification needed.

The x-ray attenuation coefficient $f$ itself could now be computed
as $e * f = R_1 * \Lambda e * f$
from the locally reconstructed $\Lambda e * f$. 
However, this is usually not done because 
the great amplitude of $\Lambda e * f$ near $f$'s discontinuities would 
substantially increase the instability of reconstructing $e * f$.
In practice, $\Lambda e * f$ also often gives enough information.






\chapter{Fourier Transforms of Singular Integrals}
\label{chap:FourierTransformsOfSingularIntegrals}

\section{Fourier Transform of the Riesz Kernel}
\label{sec:FourierTransformOfRieszKernel}


For working with the Riesz kernel $R_1$, we shall first show that
it is a tempered distribution and calculate its Fourier transform $\F R_1$,
largely according to \cite{neri71:singular}.

\begin{theorem}
  \label{thm:FourierTransformOfR1}
  Let $n \ge 2$. Then $R_1$ is a tempered distribution and
  \begin{equation}\nt
    \F R_1 (\xi) = \Fm \, |\xi|^{-1}.
  \ent\end{equation}
\end{theorem}
%
% Aineksia:
% - Neri, Prop. 2.1.1, p. 156
% - Neri, p. 152
% - Neri, Th. 4, p. 27
%

\begin{tod}
  The function $h(x) = |x|^{n-1} = \frac{1}{b_n}\,R_1(x)$ is
  locally integrable, because for any compact $K \subset B^n(0,R) \subset \Rn$,
  \begin{equation}\nt
    \begin{split}
      \int_K |x|^{1-n} \,dx & \le \int_{B^n(0,R)} |x|^{1-n}\,dx
\\&
    = |\Sm| \int_0^R r^{1-n}\, r^{n-1} \,dr = R \, |\Sm| < \infty.
    \end{split}
  \ent\end{equation}
  Write $h = g_1 + g_2$, where 
  $g_1 = h \chi_{\overline{B(0,1)}} \in L^1 \subset \Ss'$
  and $g_2 = h \chi_{\Rn \setminus \overline{B(0,1)}} \in L^2 \subset \Ss'$:
  \begin{align*}
    \nl{g_1}{1} & = |\Sm| \int_0^1 r^{1-n} \, r^{n-1} \,dr = |\Sm| < \infty,
    \\
    \nl{g_2}{2} & = |\Sm| \int_1^\infty r^{2-2n}\,r^{n-1}\,dr 
    = \frac{|\Sm|}{2-n}
    < \infty.
  \end{align*}
  Thus $h \in \Ss'$, so it makes sense to speak of 
  $\F h = \widehat{g_1} + \widehat{g_2}$, where $\widehat{g_1} \in L^\infty$
  and $\widehat{g_2} \in L^2$. 
  The distribution $\F h$ is in fact a locally integrable function, 
  since using Hölder's inequality,
  \begin{align*}
    \int_K |\F h(\xi)| \,d\xi &
    \le \int_K |\widehat{g_1}| + |\widehat{g_2}| \,d\xi
    \le \nl{\widehat{g_1}}{\infty} \,m(K) + \nl{\widehat{g_2}}{2} \, \sqrt{m(K)}
    \\ &
    \le \Fm \nl{g_1}{1} \,m(K) + \nl{g_2}{2} \, \sqrt{m(K)} < \infty.
  \end{align*}
  Because $\sigma_\lambda h(x) = |\lambda x|^{1-n} = \lambda^{1-n} h(x)$,
  \begin{equation}\nt
    \F(\sigma_\lambda h)(\xi) = \lambda^{1-n}\, \Fh(\xi).
  \ent\end{equation}
  On the other hand by Equation~\equ{FourierTransformOfDilation},
  \begin{equation}\nt
    \F(\sigma_\lambda h)(\xi) = \lambda^{-n} \sigma_{1/\lambda} \Fh(\xi),
  \ent\end{equation}
  and therefore, with $\mu = 1/\lambda$,
  \begin{equation}\nt
    \sigma_\mu \Fh = \mu^{-1} \Fh.
  \ent\end{equation}
%  \ie, $\Fh$ is homogeneous of degree $-1$ and can be written as
  Hence $\Fh$ can be written as
  \begin{equation}\nt
    \Fh(\xi) = \Fh(|\xi| {\tfrac{\xi}{|\xi|}})
    = |\xi|^{-1} \Fh({\tfrac{\xi}{|\xi|}}).
  \ent\end{equation}

  The function $h(x) = |x|^{1-n}$ is clearly radial, which is to say
  that its value at a point $x$ only depends on $|x|$.
  We shall next show that $\Fh = \widehat{g_1} + \widehat{g_2}$ is radial, 
  too, and hence 
  $\Fh(\xi/|\xi|) = C \in \C$ is a constant.
  An equivalent characterization of the radiality of a function $\phi$
  is that $\phi(Tx) = \phi(x)$ whenever $T$ is a rotation,
  \ie\  whenever $T:\Rn \to \Rn$ is a linear operator for which 
  $|\det T\,| = 1$.

  Let $T$ be a rotation. Then its transpose $T'$ and the inverse of its
  transpose ${T'}^{-1}$ are rotations, as well, since
  \begin{equation}\nt
    \det T' = \det T = 1
    \qquad \text{and} \qquad
    \det {T'}^{-1} = \frac{1}{\det T'} = \frac{1}{\det T} = 1.
  \ent\end{equation}
  The change of variable $y = T'x$, $x = {T'}^{-1}y$, 
  $dx = |\det {T'}^{-1}| \,dy = dy$ yields
  \begin{align*}
    \widehat{g_1}(T\xi) &
    = \Fm \int g_1(x)\, e^{-ix\cdot T\xi}\,dx
    = \Fm \int g_1(x)\, e^{-i(T'x)\cdot \xi} \,dx
    \\ &
    = \Fm \int g_1({T'}^{-1}y)\, e^{-iy\cdot \xi}\,dy
    = \widehat{g_1}(\xi),
  \end{align*}
  because $g_1({T'}^{-1}y) = g_1(y)$. Analogously,
  \begin{equation}\nt
  \begin{split}
    \widehat{g_2}(T\xi) 
  & = \lim_{R\to\infty} \int_{|x|\le R} g_2(x)\, e^{-ix\cdot T\xi} \,dx
  \\ &
    = \lim_{R\to\infty} \int_{|y|\le R} g_2({T'}^{-1}y)\, e^{-iy\cdot \xi} \,dy
    = \widehat{g_2}(\xi).
  \end{split}
  \ent\end{equation}
  Thus  $\Fh$ is radial and can be written as $\Fh(\xi) = C\,|\xi|^{-1}$.
  The value of the constant $C$ will be calculated using the function
  $v(x) = e^{-|x|^2/2} \in \Ss$, which has the property that $\widehat{v} = v$,
  as the following calculations show:
  \begin{equation}\nt
  \begin{split}
    \widehat{v}(\xi) 
    &
    = \Fm \int_\Rn e^{-\frac{|x|^2}{2}} e^{-ix\cdot\xi}\,dx
    \\ &
    = e^{-\frac{|\xi|^2}{2}} \,\Fm 
      \int_\Rn e^{-\frac{(x+i\xi)^2}{2}} \,dx
    \\ &
    = e^{-\frac{|\xi|^2}{2}} \,\Fm 
      \prod_{j=1}^n \infint e^{-\frac{(x_j+i\xi_j)^2}{2}} \,dx_j.
  \end{split}
  \ent\end{equation}
  For evaluating this integral, note that
  since $e^{-z^2/2}$ is analytic everywhere, its closed line
  integral along $\mathcal{C}$, the rectangle with vertices at 
  $R$, $-R$, $-R + i \xi_j$ and $R + i \xi_j$, vanishes:
  \begin{equation}\nt
  \begin{split}
    & \infint e^{-\frac{(x_j+i\xi_j)^2}{2}} \,dx_j 
    \\ &
    = \lim_{R\to\infty} \left( \oint_{\mathcal{C}} e^{-\frac{z^2}{2}}\,dz +
        \int_{-R}^R e^{-\frac{x_j^2}{2}}\,dx_j + 
        \int_0^{\xi_j} e^{-\frac{(R+it)^2}{2}} - e^{-\frac{(-R+it)^2}{2}} \,dt
        \right)
    \\ &
    = 0 + \infint e^{-\frac{x_j^2}{2}} \,dx_j + 0
    = \sqrt{2} \infint e^{-t^2} \,dt = \sqrt{2\pi}.
  \end{split}
  \ent\end{equation}
  Therefore
  \begin{equation}
    \widehat{v}(\xi) = e^{-\frac{|\xi|^2}{2}} = v(\xi).
  \end{equation}

  For calculating $C$, we evaluate $\du{h}{v}$ in two different ways.
  Firstly, using the Plancherel formula \equ{PFormulaIP}
  and the substitution $t = r^2/2$, $r = (2t)^{1/2}$, $dt = r\,dr$, we get
  \begin{equation}\nt
  \begin{split}
    \du{h(x)}{v(x)} &
    = \du{C \, |\xi|^{-1}}{v(\xi)}
    = C \,|\Sm| \int_0^\infty r^{-1} e^{-\frac{r^2}{2}}\,r^{n-1} \,dr
    \\ &
    = C \,|\Sm| \,2^{\frac{n-3}{2}} \int_0^\infty r^{\frac{n-3}{2}} e^{-t} \,dt
    = C \,|\Sm| \,2^{\frac{n-3}{2}} \Gamma({\tfrac{n-1}{2}}),
  \end{split}
  \ent\end{equation}
  and secondly,
  \begin{equation}\nt
    \du{h(x)}{v(x)}
    = |\Sm| \int_0^\infty r^{1-n}\,e^{\frac{-r^2}{2}}\,r^{n-1}\,dr
    = |\Sm|\, \sqrt{\frac{\pi}{2}}.
  \ent\end{equation}
  Hence,
  \begin{equation}\nt
    C = \frac{|\Sm|\, \pi^{1/2}\, 2^{-1/2}}{|\Sm|\, 2^{\frac{n-3}{2}}\,
                                            \Gamma(\frac{n-1}{2})}
      = \frac{\pi^{1/2}}{2^{\frac{n-2}{2}}\,\Gamma(\frac{n-1}{2})}.
  \ent\end{equation}
  Therefore, as claimed,
  \begin{equation}\nt
    \F R_1(\xi) = b_n \Fh(\xi)
    = \frac{\Gamma(\frac{n-1}{2})}{2 \pi^{\frac{n+1}{2}}} \,
      \frac{\pi^{1/2}}{2^{\frac{n-2}{2}}\,\Gamma(\frac{n-1}{2})} \,|\xi|^{-1}
    = \Fm \, |\xi|^{-1}.
  \ent\end{equation}
\end{tod}



\section{Calderón-Zygmund Theory}
\label{sec:CZTheory}

We shall next aim at justifying the formal calculation in 
\equ{LambdaInvertsConvolutionByRieszKernelFormal}
that leads to the result that the Calderón operator $\Lambda$,
which can be expressed as
\begin{equation}
  \label{CZIntroLambda}
  \Lambda = - \sum_{j=1}^n \pv \dodo{R_1}{x_j} * D_j,
\end{equation}
inverts the Riesz potential 
(Theorems \ref{thm:ExactReconstructionChap3}
and \ref{thm:LambdaAspvConvolutionChap3}).
We also want to show that when calculating the
partial derivatives of the Riesz potential,
the derivative can be taken under the integral sign,
\begin{equation}
  \label{CZIntroDjR1ConvF}
  D_j (R_1 * f) = \pv \dodo{R_1}{x_j} * f.
\end{equation}
In Equations \equ{CZIntroLambda} and \equ{CZIntroDjR1ConvF}
appears the Riesz transform, defined as follows:
\begin{definition}
  \label{def:RieszTransform}
  For $n \ge 2$, the operator
  \begin{equation}
    \pv \dodo{R_1}{x_j} \, * 
    = b_n \, (1 - n) \, \pv \frac{x_j}{|x|^{n+1}} \, *
%    : L^2(\Rn) \to L^2(\Rn)
    , \qquad j \in \{ 1, 2, \ldots, n \},
  \end{equation}
  is called the \emph{$j^{\text{th}}$ Riesz transform}.
  In one dimension,
  \begin{equation}
  \label{HilbertTransformDefinition}
    \pv \dodo{R_1}{x_1} \, * := - \pv \frac{1}{\pi x} \, * 
%    : L^2(\R) \to L^2(\R)
  \end{equation}
  is called the \emph{Riesz transform} or the \emph{Hilbert transform}.
\end{definition}

The name Hilbert transform is sometimes used instead of the Riesz transform
also in $\Rn$, $n \ge 2$.
Texts that use the notation $I_1 f$ instead of $R_1 * f$,
often write $R_j f$ for $\pv \dodo{R_1}{x_j} * f$.

Here, ``$\pv$'' stands for principal value convolution,
defined as follows:
\begin{definition}
  \label{def:pvConvolution}
  Let $n \in \Z_+$ and let $K: \Rn \to \C$ be of the form
  \begin{equation}\nt
    K(x) = \frac{\Omega(\frac{x}{|x|})}{|x|^n}.
  \ent\end{equation}
  where $\Omega: \Sm \to \C$ is an odd, bounded function.
  For $f \in L^2(\Rn)$, the \emph{principal value convolution} associated with 
  $K$ is the function
  \begin{equation}\nt
    \pv K * f(x) = \ltwolim_{\eps \to 0} \int_{|y| > \eps} K(y) f(x-y) \,dy.
  \ent\end{equation}
  The kernel $K$ is called a \emph{singular convolution kernel},
  and the operator $f \mapsto \pv K * f$ 
  a \emph{singular convolution operator}.\footnote{%
    The usual definition of a singular convolution kernel
    admits somewhat more general functions, but to simplify the treatment,
    we shall confine our study 
    to functions whose radial part is odd and bounded.
  }
\end{definition}

Observe that for $n \ge 2$, 
\begin{equation}\nt
\begin{split}
  \label{DerivativeOfRieszKernel}
  \dodo{R_1}{x_j}(x) 
  &
  = b_n\, \dodo{}{x_j} \left(\sum_{i=1}^n x_i^2\right)^{\frac{1-n}{2}} 
  \\&
  = b_n\,\frac{1-n}{2} \left(\sum_{i=1}^n x_i^2\right)^{\frac{-1-n}{2}} 2x_j^2
  \\&
  = b_n (1-n)\, \frac{x_j}{|x|^{n+1}} = b_n (1-n)\, \frac{\theta_j}{r^n}
\end{split}
\ent\end{equation}
is just the partial derivative of the Riesz kernel, but for $n = 1$,
\equ{DerivativeOfRieszKernel} does not make sense, 
so \equ{HilbertTransformDefinition} must be seen as 
the definition of $\pv \dodo{R_1}{x_1} *$.

We shall now first derive some results of the Calderón-Zygmund theory,
concerning the continuity properties of singular convolutions operators
and their Fourier transforms. 
They will be applied in Section~\sectref{sec:RieszTransforms} 
for finding answers to the questions about the Calderón operator
and the derivative of the Riesz potential.
Our derivation will largely follow \cite{neri71:singular}.

The results will mostly be needed when $n \ge 2$.
The case $n = 1$ will, be required for two-dimensional computed tomography,
since the proof of Theorem~\ref{thm:SKthm51} uses 
the Calderón-Zygmund theory results in $\R^{n-1}$.

Many of the following facts would be simpler to derive in one dimension.
As the main focus is on $n \ge 2$, 
we shall not treat that situation separately, 
but only consider general $n \in \Z_+$,
which covers $n = 1$ as a special case.


Our first result is that 
the Fourier transform of a truncated convolution kernel 
is uniformly bounded, and has a pointwise limit 
as the truncation radii are taken to zero and infinity:

\begin{lemma}
% Neri, Lemma 3, page 93
  \label{lemma:KHatExists}
  For $n \in \Z_+$, let $\Omega: \Sm \to \C$ be an odd, bounded function and
  \begin{equation}
    K(x) = \frac{\Omega(\frac{x}{|x|})}{|x|^n}, \qquad \qquad
    K_{\eps,\eta} (x) = \begin{cases}
                           K(x), & \eps < |x| < \eta \\
                           0,    & otherwise.
                         \end{cases}
  \end{equation}
  Then there is a constant $A > 0$ such that
  \begin{equation}\nt
    |\widehat{K}_{\eps,\eta}(\xi)| \le A
    \qquad 
    \text{for all } \xi \in \Rn \setminus \{0\} 
    \text{ and all } \eps, \eta > 0,
  \ent\end{equation}
  and the limit
  \begin{equation}\nt
    \widehat{K}(\xi) 
    = \lim_{\substack{\eps \to 0 \\ \eta \to \infty}} 
      \widehat{K}_{\eps,\eta}(\xi)
  \ent\end{equation}
  exists for all $\xi \in \Rn \setminus \{ 0 \}$.
\end{lemma}

\begin{tod}
  Fix $\xi \in \Rn \setminus \{0\}$ and write $\rho = |\xi|$.
  Introducing spherical coordinates, with $\varphi$ measuring the angle
  between $x$ and $\xi$, yields for $x \ne 0$
  \begin{equation}\nt
    r = |x|, \qquad
    \theta = \frac{x}{|x|}, \qquad 
    dx = r^{n-1}\,dr\,d\theta, \qquad
    x \cdot \xi = r \rho \cos \varphi, 
  \ent\end{equation}
  and
  \begin{equation}\nt
  \begin{split}
    \widehat{K}_{\eps,\eta}(\xi) 
    & = \Fm \int_{\eps<|x|<\eta} K(x) \, e^{-ix \cdot \xi} \, dx
    \\&
    = \Fm \int_\Sm \Omega(\theta) \int_\eps^\eta e^{-ir\rho \cos\varphi}
      \, \frac{dr}{r} \,d\theta.
  \end{split}
  \ent\end{equation}
  In the inner integral, make the substitution $s = \rho r$, $ds = \rho\,dr$
  to get
  \begin{equation}\nt
    \widehat{K}_{\eps,\eta}(\xi)
    = \Fm \int_{\Sm} \Omega(\theta)
      \int_{\eps \rho}^{\eta \rho} e^{-is \cos\varphi} \, \frac{ds}{s} 
      \, d\theta.
  \ent\end{equation}
  Since $\Omega$ is an odd function,
  \begin{equation}
    \label{MeanValueOfOmegaIsZero}
    \int_\Sm \Omega(\theta) \,d\theta = 0
  \end{equation}
  and
  \begin{equation}\nt
    \int_\Sm \Omega(\theta) \,d\theta \int_\eps^\eta \chi_{(0,1)}(s)\,ds = 0.
  \ent\end{equation}
  Thus,
  \begin{equation}\nt
    \widehat{K}_{\eps,\eta}(\xi)
    = \Fm \int_{\Sm} \Omega(\theta) \, I_{\eps,\eta} \, d\theta,
  \ent\end{equation}
  where
  \begin{equation}\nt
    I_{\eps,\eta}(\theta) = \int_{\eps \rho}^{\eta \rho} 
      \frac{e^{-is \cos\varphi} - \chi_{(0,1)}(s)}{s} \, ds 
  \ent\end{equation}
  It now suffices to find a function $g \in L^1(\Sm)$ 
  that dominates the absolute value of the inner integral $I_{\eps,\eta}$:
  \begin{equation}
    \label{NeriLemmaThreeGDominates}
    |I_{\eps,\eta}(\theta)| \le g(\theta)
    \qquad  \text{for all } \eps,\, \eta \ge 0.
  \end{equation}
  In fact, if we find such a $g$, we immediately get the estimate
  \begin{equation}\nt
    |\widehat{K}_{\eps,\eta}(\xi)| \le \Fm \sup_{\theta\in\Sm} |\Omega(\theta)|
    \int_\Sm g(\theta) \, d\theta =: A,
  \ent\end{equation}
  proving the first part of the claim, and the second part follows
  from Lebesgue's theorem of dominated convergence:
  As $\eps\to 0$ and $\eta \to \infty$,
  \begin{equation}\nt
    \Omega(\theta) I_{\eps,\eta}(\theta) 
    \to \Omega(\theta) I_{0,\infty}(\theta)
    \quad \text{pointwise}
  \ent\end{equation}
  and $|\Omega(\theta) I_{\eps,\eta}(\theta)| \le |\Omega(\theta)|\, g(\theta)
  \in L^1(\Sm)$, whence
  \begin{equation}\nt
    \widehat{K}_{\eps,\eta}(\xi)
    = \int_\Sm \Omega(\theta) I_{\eps,\eta}(\theta) \,d\theta
    \to \int_\Sm \Omega(\theta) I_{0,\infty}(\theta) \,d\theta,
  \ent\end{equation}
  which is finite.

  We now choose
  \begin{equation}\nt
    g(\theta) = -2 \ln |\cos\varphi \,| + C, 
  \ent\end{equation}
  where $C < \infty$ is a constant,
  and show that $g \in L^1(\Sm)$ and that \equ{NeriLemmaThreeGDominates}
  holds with an appropriate choice of $C$.

  Since changing the sign of $\cos \varphi$ only changes $I_{\eps,\eta}(\xi)$
  to its complex conjugate, we may assume that $\cos \varphi > 0$.
  We prove \equ{NeriLemmaThreeGDominates} separately in the three cases
  \begin{enumerate}
    \item $\eps \rho \le 1 \le \eta \rho$, \label{NeriLemmaThreeCaseOne}
    \item $\eps \rho > 1$,  \label{NeriLemmaThreeCaseTwo}
    \item $\eta \rho < 1$.  \label{NeriLemmaThreeCaseThree}
  \end{enumerate}
  In case~\ref{NeriLemmaThreeCaseOne}, $I_{\eps,\eta}(\theta) = I_1 + I_2$,
  where
  \begin{equation}\nt
    I_1 = \int_{\eps \rho}^1 \frac{e^{-is\cos\varphi}-1}{s} \,ds
    \qquad \text{and} \qquad
    I_2 = \int_1^{\eta \rho} \frac{e^{-is\cos\varphi}}{s} \,ds.
  \ent\end{equation}
  Because $|e^{iu}-1| = |e^{iu/2}(e^{iu/2}-e^{-iu/2})| 
  = |2i\sin\frac{u}{2}| \le |u|$, 
  \begin{equation}\nt
    |I_1| \le \int_{\eps \rho}^1 \frac{s \cos\varphi}{s} \,ds \le 1.
  \ent\end{equation}
  The substitution $t = s \cos\varphi$, $dt = ds \cos\varphi$ gives us 
  \begin{equation}\nt
  \begin{split}
    |I_2| 
  & = |\int_{\cos\varphi}^{\eta \rho \cos\varphi} \frac{e^{-it}}{t} \,dt|
  \\ &
    \le \int_{\cos\varphi}^1 \frac{dt}{t}
    + \left|\int_1^{\eta \rho \cos\varphi} \frac{e^{-it}}{t} \,dt \right|
    \le - \ln \cos \varphi + C_1,
  \end{split}
  \ent\end{equation}
  where
  \begin{equation}\nt
    C_1 = \sup_{v > 1} \left| \int_1^v \frac{e^{-iu}}{u} \,du \right|
  \ent\end{equation}
  is finite, since it is the supremum of a continuous function defined 
  on $[1,\infty)$ whose limit at infinity is finite:
  \begin{equation}\nt
  \begin{split}
    \left| \int_1^\infty \frac{e^{-iu}}{u} \,du \right|
    & = \left| i \sij{1}{\infty} \frac{e^{-iu}}{u} 
    + i \int_1^\infty \frac{e^{-iu}}{u^2} \,du \right|
    \\ &
    \le |-ie^{-i}| + \int_1^\infty \frac{du}{u^2} < \infty.
  \end{split}
  \ent\end{equation}
  Thus in case~\ref{NeriLemmaThreeCaseOne}, 
  $|I_{\eps,\eta}(\theta)| \le |I_1| + |I_2| \le - \ln \cos \varphi + C_1 + 1$.

  In case~\ref{NeriLemmaThreeCaseTwo}, $I_{\eps,\eta}(\theta) = I_2 - I_3$,
  where
  \begin{equation}\nt
    I_3 = \int_1^{\eps \rho} \frac{e^{-is\cos\varphi}}{s}\,ds,
  \ent\end{equation}
  which is of the same form as $I_2$ above, with $\eta$ replaced by $\eps$.
  Thus, we get the same estimate for $I_3$ as for $I_2$, and
  $|I_{\eps,\eta}(\theta)| \le |I_2| + |I_3| \le -2 \ln \cos \varphi + 2 C_1$.

  In case~\ref{NeriLemmaThreeCaseThree},
  \begin{equation}\nt
  \begin{split}
    |I_{\eps,\eta}(\theta)| 
    &
    \le \int_{\eps \rho}^{\eta \rho} 
        \left|\frac{e^{-is\cos\varphi}-1}{s} \right| \,ds
    \\&
    \le \int_0^1 \left| \frac{e^{-is\cos\varphi}-1}{s} \right| \,ds
    \le \int_0^1 \frac{s\cos\varphi}{s} \,ds \le 1.
  \end{split}
  \ent\end{equation}
  Therefore with $C = 2C_1 + 1$, 
  \begin{equation}\nt
    |I_{\eps,\eta}(\theta)| \le - \ln \cos \varphi + C = g(\theta)
  \ent\end{equation}
  in all three cases, proving \equ{NeriLemmaThreeGDominates}.
  The fact that $g \in L^1(\Sm)$ can easily be verified 
  using the spherical coordinates of 
  Lemma~\ref{lemma:SphericalCoordinatesSm}: If $n > 2$,
  \begin{equation}\nt
  \begin{split}
    \int_\Sm - \ln |\cos \varphi \,| \,d\theta 
    &
    = - 2 \int_\spxi \ln \cos \varphi \,d\theta
    \\ &
    = - 2\, |S^{n-2}| \int_0^{\pi/2} \ln\cos\varphi \, \sin^{n-2}\varphi 
      \,d\varphi
    \\ &
    \le 2\, |S^{n-2}| \int_0^{\pi/2} -\sin\varphi \, \ln\cos\varphi \,d\varphi
    \\ &
    = 2 \, |S^{n-2}| \sij{0}{\pi/2} \cos\varphi \, \ln\cos\varphi - \cos\varphi
    \\ &
    = 2 \, |S^{n-2}| < \infty,
  \end{split}
  \ent\end{equation}
  and if $n = 2$,
  \begin{equation}\nt
    \int_{S^1} - \ln |\cos \varphi \,| \,d\theta 
    = -4 \int_0^{\pi/2} \frac{\varphi^2}{2} + \varphi^4 \, b(\varphi) 
      \,d\varphi
    < \infty,
  \ent\end{equation}
  where $b$ is a bounded function.

  The case $n = 1$ would be quite simple to handle all the way on its own,
  but this would be unnecessary, as the treatment above covers it as well.
  Now $S^0 = \{-1, +1\}$ and
  \begin{equation}
    \int_{S^0} f(x) \,dx = f(-1) + f(1).
  \end{equation}
  The angle between $x$ and $\xi$ is either $0$ or $\pi$, 
  and $g(\pm 1) = \ln |\mspace{-2.5mu}\pm 1\mspace{2mu}| + C = C$, so
  $\norm[L^1(S^0)]{g} = C + C < \infty$.
\end{tod}


Not only do we know that $\FK$ exists, we know how to calculate it:

\begin{lemma}
  % Neri, Theorem 4, page 98, sightly modified
  \label{lemma:FormulaForFK}
  Let $n \in \Z_+$ and let $K$ be of the form
  \begin{equation}\nt
    K(x) = \frac{\Omega(\frac{x}{|x|})}{|x|^n},
  \ent\end{equation}
  where $\Omega:\Sm \to \C$ is an odd, bounded function,
  and let $\FK$ be as in Lemma~\ref{lemma:KHatExists}. 
  Then for all $\xi \neq 0$,
  \begin{equation}
    \label{FormulaForFK}
    \FK(\xi) = - i \pi\, \Fm\, \int_\spxi \Omega(\theta) \, d\theta.
  \end{equation}
\end{lemma}

\begin{tod}
  We use the same notation as in the proof of Lemma~\ref{lemma:KHatExists}:
  \begin{equation}\nt
    \rho = |\xi|, \quad
    r = |x|, \quad
    \theta = \frac{x}{|x|}, \quad 
    dx = r^{n-1}\,dr\,d\theta, \quad
    x \cdot \xi = r \rho \cos \varphi
  \ent\end{equation}
  and
  \begin{equation}\nt
    \widehat{K}_{\eps,\eta}(\xi) = \Fm \int_\Sm \Omega(\theta)
      \int_{\eps \rho}^{\eta \rho} \frac{e^{-is \cos\varphi}}{s} \,ds\,d\theta.
  \ent\end{equation}
  Lemma~\ref{lemma:KHatExists} shows that when $\xi \neq 0$, the limit
  \begin{equation}\nt
    \widehat{K}(\xi) = \lim_{\delta \to 0} \Fm \int_\Sm \Omega(\theta) 
      \int_\delta^\infty \frac{e^{-is \cos\varphi}}{s} \,ds \,d\theta
  \ent\end{equation}
  exists. We now use the fact that $\Omega$ is odd and note that
  if we think of $\varphi\in [0,\pi]$ as a function of $\theta \in \Sm$,
  $\cos \varphi(-\theta) = \cos (\pi-\varphi(\theta)) = - \cos \varphi(\theta)$.
  Hence
  \begin{equation}\nt
  \begin{split}
    \int_\Sm \Omega (\theta) \int_\delta^\infty
      \frac{e^{-is \cos\varphi}}{s} \,ds \,d\theta
    &
    = \int_\spxi \Omega(\theta) \int_\delta^\infty 
      \frac{e^{-is \cos\varphi} - e^{is \cos \varphi}}{s} \,ds \,d\theta 
    \\ &
    = \int_\spxi \Omega(\theta) \int_\delta^\infty 
      \frac{-2i \sin(s \cos\varphi)}{s} \,ds \,d\theta.
  \end{split}
  \ent\end{equation}
  The change of variable $t = 2\pi s \cos\varphi$,
  $dt = 2 \pi \cos\varphi \,ds$ leads to
  \begin{equation}\nt
  \begin{split}
    \widehat{K}(\xi)
  & = - 2i \,\Fm\, \lim_{\delta \to 0}
      \int_\spxi \Omega(\theta) \int_\delta^\infty \frac{\sin t}{t} 
      \,dt \,d\theta
  \\ &
    = -i \pi \,\Fm \int_\spxi \Omega(\theta) \,d\theta,
  \end{split}
  \ent\end{equation}
  as claimed, since the inner integral evaluates to $\pi/2$, 
  which can be seen as follows: If $g \in L^2(\R)$ is the triangular function
  $g(x) = \max(0, 1-|x|/2)$, then
  \begin{equation}\nt
  \begin{split}
    \sqrt{2\pi} \, \widehat{g}(\xi) &
    = \int_{-2}^0 (1+{\tfrac{x}{2}})\, e^{-ix\xi} \,dx 
      + \int_0^2 (1-{\tfrac{x}{2}})\, e^{-ix\xi} \,dx
  \\ &
    = \int_0^2 (1-{\tfrac{y}{2}})\, e^{iy\xi} \,dy
      + \int_0^2 ({1-\tfrac{x}{2}})\, e^{-ix\xi} \,dx
  \\ &
    = \int_0^2 (1-{\tfrac{x}{2}})\, 2 \cos x\xi \,dx
  \\ &
    = 2 \sij{0}{2} \frac{\sin x\xi}{\xi}
      - \sij{0}{2} \frac{x \sin{x\xi}}{\xi}
      + \int_0^2 \frac{\sin x\xi}{\xi} \,dx
  \\ &
    = \frac{1 - \cos 2\xi}{\xi^2},
  \end{split}
  \ent\end{equation}
  and the inverse Fourier transform gives
  \begin{equation}
  \begin{split}
    \frac{\pi}{2} 
    &
    = \frac{\pi}{2}\, g(0)
    \\&
    = \frac{\pi}{2} \frac{1}{\sqrt{2\pi}} \infint 
        \widehat{g}(\xi) \, e^{i 0 \xi} \,d\xi
    \\&
    = \frac{\pi}{2} \frac{1}{\sqrt{2\pi}} 
      \infint \frac{1}{\sqrt{2\pi}} \frac{1 - \cos 2\xi}{\xi^2} \,d\xi
    \\&
    = - \frac{1}{4} \sij{-\infty}{\infty} \frac{1 - \cos 2\xi}{\xi}
      + \frac{1}{2} \infint \frac{\sin 2\xi}{\xi} \,d\xi
    \\&
    = 0 + \frac{1}{2} \infint \frac{\sin x}{x} \,dx
    \\&
    = \int_0^\infty \frac{\sin x}{x}\,dx.
  \end{split}
  \end{equation}
\end{tod}

Aiming at establishing continuity results for
the operator $\pv K *: L^2 \to L^2$
and showing that a certain form of 
the convolution theorem \equ{ConvolutionTheorem} holds, 
we shall first prove the existence of the convolution 
$K_{\eps,\infty} * f$ truncated from below:

\begin{lemma}
  \label{lemma:TruncatedConvolutionWithKExists}
  Assume that $n \in \Z_+$, $f \in L^2(\Rn)$, $\Omega \in L^1(\Sm)$ and
  let $K$ and $K_{\eps,\eta}$ be as in Lemma~\ref{lemma:KHatExists}.
  Then 
  \begin{equation}\nt
    K_{\eps,\infty} * f (x) = \int_{|y| > \eps} K(y) f(x-y) \,dy
  \ent\end{equation}
  exists almost everywhere.
\end{lemma}

\begin{tod}
  Because
  \begin{equation}\nt
    |K_{\eps,\infty} * f (x)|
    \le \int_{|y| > \eps} |K(y)| |f(x-y)| \,dy =: I(x),
  \ent\end{equation}
  it suffices to show that $I(x)$ is locally integrable, because
  in that case it must be finite almost everywhere.
  Let $B \subset B(0,R_B) \subset \Rn$ be compact. Fubini's theorem yields
  \begin{equation}\nt
  \begin{split}
    \int_B |I(x)| \,dx
  & = \int_B \int_{|y| > \eps} 
      \frac{|\Omega(\frac{y}{|y|})|}{|y|^n} \, |f(x-y)| \,dy \,dx
  \\ &
    = \int_B \int_{\Sm} |\Omega(\theta)| 
      \int_\eps^\infty |f(x - r\theta)| \,r^{-1} \,dr \,d\theta \,dx
  \\ &
    = \int_\Sm |\Omega(\theta)| 
      \int_B \int_\eps^\infty |f(x - r\theta)| \,r^{-1} \,dr \,dx \,d\theta
  \end{split}
  \ent\end{equation}
  since the integrand is non-negative. We use Hölder's inequality twice,
  first in the innermost integral to get
  \begin{equation}\nt
  \begin{split}
    \int_\eps^\infty |f(x - r\theta)| \,r^{-1} \,dr
  & \le \sqrt{\int_\eps^\infty r^{-2} \,dr} 
        \sqrt{\int_\eps^\infty |f(x - r\theta)|^2 \,dr}
    \\&
    = \frac{1}{\sqrt{\eps}} \sqrt{\int_\eps^\infty |f(x - r\theta)|^2 \,dr}
  \end{split}
  \ent\end{equation}
  and then in the middle integral to get
  \begin{equation}\nt
  \begin{split}
    \int_B \int_\eps^\infty |f(x - r\theta)| \,r^{-1} \,dr \,dx
  & \le \frac{1}{\sqrt{\eps}} 
        \int_B \sqrt{\int_\eps^\infty |f(x - r\theta)|^2 \,dr} \,dx
    \\ &
    \le \sqrt{\frac{m(B)}{\eps}
              \int_B \int_\eps^\infty |f(x - r\theta)|^2 \,dr \,dx}
    \\ &
    \le \sqrt{\frac{m(B)}{\eps} \int_B \infint |f(x - r\theta)|^2 \,dr \,dx}.
  \end{split}
  \ent\end{equation}
  Because 
  $B \subset \{ x' + t\theta \in \Rn \,|\, x' \cdot \theta = 0,\, |t|<R_B \}$,
  the change of variable $x = x' + t\theta$, $dx = dx'\,dt$ gives us
  \begin{equation}\nt
  \begin{split}
  & \sqrt{\frac{m(B)}{\eps} \int_B \infint |f(x - r\theta)|^2 \,dr \,dx}
    \\ &
    \le \sqrt{\frac{m(B)}{\eps}
              \int_{-R_B}^{R_B} \int_{\theta^\bot} \infint
              |f(x' + t\theta - r\theta)|^2 \,dr \,dx' \,dt}
    \\ &
    = \sqrt{\frac{m(B)}{\eps} \int_{-R_B}^{R_B} \nl{f}{2}^2 \,dt}
    = \sqrt{\frac{2 R_B\, m(B)}{\eps}}\, \nl{f}{2},
  \end{split}
  \ent\end{equation}
  whence we conclude that
  \begin{equation}\nt
    \int_B |I(x)| \,dx 
    \le \nl{\Omega}{1} \, \sqrt{\frac{2 R_B \, m(B)}{\eps}} \, \nl{f}{2}
    < \infty,
  \ent\end{equation}
  as claimed.
\end{tod}


We are now ready to show that $\pv K*$ is a continuous operator on $L^2$:

\begin{theorem}[Calderón-Zygmund]
% Neri, Theorem 3, page 97 (weaker form)
\samepage
  \label{thm:CalderonZygmund}
  Assume that $n \in \Z_+$,
  let $\Omega: \Sm \to \C$ be an odd, bounded function and let
  \begin{equation}
    K(x) = \frac{\Omega(\frac{x}{|x|})}{|x|^n}, \qquad \qquad
    K_{\eps,\eta} (x) = \begin{cases}
                           K(x), & \eps < |x| < \eta \\
                           0,    & \text{otherwise}
                         \end{cases}
  \end{equation}
  and
  \begin{equation}\nt
    \widehat{K}(\xi) 
    = \lim_{\substack{\eps \to 0 \\ \eta \to \infty}} 
      \widehat{K}_{\eps,\eta}(\xi).
  \ent\end{equation}
  Then for all $f \in L^2(\Rn)$,
  \begin{enumerate}
      \item[(a)] there exists a constant $B > 0$, 
        independent of $\eps$, $\eta$, and $f$, such that 
        \begin{equation}
          \nl{K_{\eps,\eta} * f}{2} \le B \nl{f}{2}
        \end{equation}
      \item[(b)] the limit
        \begin{equation}
          \label{CZTheoremFTildeDef}
          \tilde{f} 
          := \ltwolim_{\substack{\eps\to 0 \\ \eta \to \infty}} 
             K_{\eps,\eta} * f = \pv K * f
        \ent\end{equation}
        exists
      \item[(c)]
        the Fourier transform of $\tilde{f}$ satisfies
        \begin{equation}
          \F(\tilde{f}) = \Fp\, \widehat{K} \Ff
          \qquad \text{and} \qquad
          \nl{\tilde{f}}{2} \le M \, \nl{f}{2},
        \end{equation}
        where 
        $M = \Fp\, \sup_{\xi\in\Rn \setminus \{0\}} |\widehat{K}(\xi)| <\infty$
      \item[(d)] if $f$ is in addition Hölder-$\alpha$-continuous
        for some $\alpha > 0$, \ie\ for all $x \in \Rn$
        there exist constants $\delta, C > 0$ such that 
        \begin{equation}\nt
          |f(x) - f(y)| < C\, |x-y|^\alpha \quad 
          \text{whenever} \quad |x-y| \le \delta,
        \ent\end{equation}
        then the pointwise limit
        \begin{equation}\nt
          \lim_{\substack{\eps \to 0 \\ \eta \to \infty}}
            K_{\eps,\eta} * f(x) 
        \ent\end{equation}
        exists for all $x \in \Rn$ 
        and it equals $\tilde{f}(x)$ almost everywhere.
  \end{enumerate}
\end{theorem}

\begin{tod}
  Lemma~\ref{lemma:KHatExists} asserts that $|\widehat{K}_{\eps,\eta}| \le A$
  for some fixed $A > 0$, whence the Parseval formula~\equ{PFormulaNorm}
  and the Convolution theorem \equ{ConvolutionTheorem} imply part (a):
  \begin{equation}\nt
  \begin{split}
    \nl{K_{\eps,\eta} * f}{2}
  & = \nl{\F(K_{\eps,\eta} * f)}{2}
    = \Fp \, \nl{\widehat{K}_{\eps,\eta} \Ff}{2} 
    \\&
    \le \Fp \, A \, \nl{\Ff}{2}
    = B\, \nl{f}{2}
  \end{split}
  \ent\end{equation}
  with $B = \Fp\, A$.
  If we set $\tilde{f} = \Fi\big(\Fp \, \FK \Ff\big)$, 
  part (c) is obviously true, since
  \begin{equation}\nt
    \nl{\tilde{f}}{2} = \nl{\F(\tilde{f})}{2} 
    = \Fp\, \nl{\widehat{K} \Ff}{2}
    \le M \nl{\Ff}{2}
    = M \nl{f}{2}.
  \ent\end{equation}
  where 
  $M = \Fp\,\sup_{\xi\in\Rn \setminus \{0\}} |\widehat{K}(\xi)| \le B <\infty$.
  This $\tilde{f}$ satisfies (b), too, since
  \begin{equation}\nt
  \begin{split}
    \nl{K_{\eps,\eta} * f - \tilde{f}}{2}
  & = \nl{\F(K_{\eps,\eta} * f - \tilde{f})}{2}
    \\&
    = \nl{(\widehat{K}_{\eps,\eta} - \widehat{K}) \Ff}{2}
    \xrightarrow{\eps \to 0,\, \eta \to \infty} 0,
  \end{split}
  \ent\end{equation}
  as Lebesgue's theorem of dominated convergence implies:
  Because $|\widehat{K}_{\eps,\eta} - \widehat{K}| \le 2A$,
  \begin{equation}\nt
    |(\widehat{K}_{\eps,\eta} - \widehat{K})^2 \Ff^2| \le 4A^2 |\Ff|^2
  \ent\end{equation}
  which is in $L^1$ since $\nl{4A^2 \, |\Ff|^2}{1} 
  = 4A^2 \nl{\Ff}{2}^2 = 4A^2 \nl{f}{2}^2 < \infty$.
  As, in addition, 
  \begin{equation}\nt
    (\widehat{K}_{\eps,\eta}(\xi) - \widehat{K}(\xi))^2 \Ff(\xi)^2
    \to 0 \qquad \text{for almost all } \xi
  \ent\end{equation}
  by Lemma~\ref{lemma:KHatExists}, Lebesgue's theorem asserts that
  \begin{equation}\nt
    \nl{(\widehat{K}_{\eps,\eta} - \widehat{K}) \Ff}{2}^2
    = \nl{(\widehat{K}_{\eps,\eta} - \widehat{K})^2 \Ff^2}{1}
    \xrightarrow{\eps \to 0,\, \eta \to \infty} \nl{0}{1} = 0.
  \ent\end{equation}
  Part (d) remains to be proved. Fix $x \ne 0$ and let $0 < \eps < \delta$. 
  By Lemma~\ref{lemma:TruncatedConvolutionWithKExists}, 
  $K_{\delta,\infty} * f$ exists. Because $\Omega$ is odd and bounded,
  \begin{equation}\nt
    \int_{\eps \le |y| \le \delta} \frac{\Omega(\frac{y}{|y|})}{|y|^n} \,f(x)
      \, dy
    = f(x) \int_\eps^\delta \frac{dr}{r} \int_\Sm \Omega(\theta) \,d\theta
    = 0,
  \ent\end{equation}
  whence
  \begin{equation}\nt
  \begin{split}
    \left| K_{\eps,\infty} * f(x) - K_{\delta,\infty} * f(x) \right|
    &
    = \Big| \int_{\eps \le |y| \le \delta} 
      \frac{\Omega(\pfrac{y}{|y|})}{|y|^n} \, f(x - y) \,dy \,\Big|
    \\&
    = \Big| \int_{\eps \le |y| \le \delta} 
      \frac{\Omega(\pfrac{y}{|y|})}{|y|^n} \, \big(f(x - y) - f(x)\big) 
      \, dy \,\Big|
    \\&
    \le \int_{\eps \le |y| \le \delta} 
      \frac{|\Omega(\pfrac{y}{|y|})|}{|y|^n} \, \big|f(x - y) - f(x)\big| \,dy.
  \end{split}
  \ent\end{equation}
  By the assumption about Hölder continuity, 
  $|f(x - y) - f(x)| \le C \, |y|^\alpha$ and therefore
  \begin{equation}\nt
  \begin{split}
    \left| K_{\eps,\infty} * f(x) - K_{\delta,\infty} * f(x) \right|
    &
    \le \int_\eps^\delta \int_\Sm |\Omega(\theta)| \, d\theta \,
      r^{-n} \, C r^{\alpha} \, r^{n-1} \,dr
    \\&
    \le C \, \nl{\Omega}{\infty} \, |\Sm| 
      \,\frac{\delta^\alpha - \eps^\alpha}{\alpha}
    \\&
    \le C \, \nl{\Omega}{\infty} \, |\Sm| \,\frac{\delta^\alpha}{\alpha}
    \xrightarrow{\delta \to 0} 0.
  \end{split}
  \ent\end{equation}
  This shows that if $\delta_k \xrightarrow{k \to \infty} 0$, 
  then $\big(K_{\delta_k,\infty} * f(x)\big)_{k \in \N}$ is a Cauchy sequence,
  so that the limit
  \begin{equation}
    \lim_{k \to \infty} K_{\delta_k, \infty} * f(x)
  \end{equation}
  exists. 
  Since this is true for all sequences $(\delta_k)$ converging to $0$,
  this limit is also the limit $\lim_{\delta \to 0} K_{\delta, \infty} * f(x)$.
  Therefore the pointwise limit exists.

  Now if $\eps_k \to 0$,
  $\eta_k \to \infty$ and $f_k = K_{\eps_k,\eta_k} * f$,
  the sequence $f_k$ converges to $\tilde{f}$ in $L^2$.
  Hence there exists a subsequence $(f_{k_j})$  such that
  for almost all $x$, $f_{k_j}(x) \to \tilde{f}(x)$.
  On the other hand, since the pointwise limit $\lim_{k\to\infty} f_k(x)$
  exists, it must be equal to the limit of the subsequence. 
  Consequently
  \begin{equation}\nt
    \lim_{k \to \infty} f_k(x) = \lim_{j \to \infty} f_{k_j}(x) 
    = \tilde{f}(x)
  \ent\end{equation}
  for almost all $x$.
\end{tod}


It is also true that the pointwise limit
\begin{equation}
  \pv K * f(x) = \lim_{\eps \to 0} \int_{|y| > \eps} K(y) \, f(x - y) \,dy
\end{equation}
exists for almost all $x \in \Rn$ 
even without the condition of Hölder continuity,
and under somewhat looser assumptions about $\Omega$.
Since we shall not need this result, we only refer to
\cite[Theorem I.IV.3.8, page 113]{neri71:singular}.

The function $\tilde{f}$ is defined above in \equ{CZTheoremFTildeDef}
as a principal value limit of truncated convolutions.
It would be pleasant to be able to ``pass the limit under the integral'',
\ie, to view $\tilde{f}$ as a convolution of $f$ with some distribution.
The following theorem does this for $f \in \DD$.

\begin{theorem}
  \label{thm:VPKIsADistribution}
  Let $n \in \Z_+$, 
  let $\Omega:\Sm \to \C$ be an odd, bounded function and let $K$, 
  $K_{\eps,\eta}$ and $\FK$ be as in Theorem~\ref{thm:CalderonZygmund}.
  Then $\pv K$, defined by
  \begin{equation}\nt
    \du{\pv K}{\phi} = \lim_{\eps\to 0} \int_{|x|>\eps} K(x) \phi(x) \,dx,
  \ent\end{equation}
  is a tempered distribution and $\F(\pv K) = \FK \in L^\infty(\Rn)$.
  Furthermore for $f \in \DD$,
  \begin{equation}
    \label{LTwoLimEqualspvKConvf}
    \tilde{f} := \ltwolim_{\substack{\eps\to 0 \\ \eta \to \infty}}
                          K_{\eps,\eta} * f = (\pv K) * f
  \end{equation}
  and
  \begin{equation}
    \label{FpvKConvfEqualsFpFpvKFf}
    \F((\pv K) * f) = \Fp \, \F(\pv K)\, \F(f).
  \end{equation}
\end{theorem}
% Neri, Th. 2.1.6, 2.1.5 ja omaa

\begin{tod}
  We begin by showing that $\pv K$ is a distribution:
  We shall first show that for all $\phi \in \DD$, 
  $\du{\pv K}{\phi} = I_1 + I_2$ is finite, where
  \begin{equation}\nt
    I_1 = \lim_{\eps\to 0} \int_{\eps < |x| < 1} K(x) \phi(x) \,dx,
    \qquad
    I_2 = \int_{|x| \ge 1} K(x) \phi(x)\,dx.
  \ent\end{equation}
  Since $\Omega$ is odd,
  \begin{equation}\nt
    \int_{\eps < |x| < 1} K(x) \phi(0) \, dx
    = \phi(0) \int_{\Sm} \Omega(\theta) \,d\theta \int_\eps^1 \frac{dr}{r} = 0
  \ent\end{equation}
  and hence we can write
  \begin{equation}\nt
    I_1 = \lim_{\eps\to 0} \int_{\eps < |x| < 1} g(x) \,dx,
  \ent\end{equation}
  where $g(x) = K(x) (\phi(x) - \phi(0)) = K(x) \nabla \phi(tx) \cdot x$
  for some $t \in [0,1]$. Now, if 
  \begin{equation}\nt
    C = \sup_{|t| \le 1} |\nabla \phi (tx)| < \infty
    \qquad \text{and} \qquad
    M_0 = \sup_{\theta \in \Sm} |\Omega(\theta)|,
  \ent\end{equation}
  we see that 
  $|g(x)| \le |\Omega(\theta)|\,|x|^{-n} \, |\nabla \phi (tx)|\, |x|
  \le C M_0 \,|x|^{1-n}$ and hence
  \begin{equation}\nt
    |I_1| \le C M_0 \int_{\Sm} d\theta \int_0^1 r^{1-n} r^{n-1}\,dr
    = C M_0 \,|\Sm| < \infty.
  \ent\end{equation}
  Since also
  \begin{equation}\nt
    |I_2| \le \int_{|x| \ge 1} |K(x) \phi(x)| \,dx
    \le \sup_{|x| \ge 1} |K(x)| \, \nl{\phi}{1} < \infty,
  \ent\end{equation}
  we conclude that $\du{\pv K}{\phi}$ is finite for all $\phi \in \DD$.

  Since $\pv K$ is clearly linear, it remains to show that it is continuous.
  Let $\phi_k \to \phi$ in $\DD$. By the definition of convergence in $\DD$,
  there exists an $R>0$ such that for all $k \in N$, 
  $\supp \phi_k \subset \closure{B(0,R)}$, and 
  \begin{equation}\nt
    \sup_{|x| \le R} |\nabla (\phi_k(x) - \phi(x))|
    \le \sqrt{\sum_{j=1}^n \sup_{|x| \le R}
      \Big| \dodo{\phi_k}{x_j}(x) - \dodo{\phi}{x_j}(x) \Big|^2}
    \xrightarrow{k\to\infty} 0.
  \ent\end{equation}
  Hence as above, for some $t \in [0,1]$,
  \begin{align*}
    |\du{\pv K}{\phi_k - \phi}|
  & = \Big| \lim_{\eps\to 0}
      \int_{|x| > \eps} \frac{\Omega(\frac{x}{|x|})}{|x|^n}
      \big((\phi_k(x) - \phi(x)) - (\phi_k(0) - \phi(0))\big) \,dx \, \Big|
  \\ &
    \le \lim_{\eps\to 0} \int_\eps^R \int_{\Sm} \frac{|\Omega(\theta)|}{r^n} \,
      |\nabla (\phi_k - \phi)(tx)|\,|x| \, r^{n-1} \, dr \, d\theta
  \\ &
    \le M_0 R\,|\Sm|\,\sup_{|x| \le R} |\nabla(\phi_k - \phi)(x)|
    \xrightarrow{k\to\infty}0,
  \end{align*}
  which shows that $\pv K$ is continuous and consequently a distribution.

  To show that this distribution is tempered, first consider the distribution
  $u =
  \linebreak
  (1 + |x|^2)^{-1} (\pv K)$.
  Let $\zeta \in \DD$ be such that $\zeta(x) = 1$ for $|x| \le 1$;
  for instance the function $\chim_{1,2}$ of 
  \equ{ApproxCharacteristicFunction}.
  Write $u_1 = \zeta u$, $u_2 = u - u_1$.
  Now $u_1 \in \Ss'$, which is seen as follows: If $\phi \in \Ss$,
  $\zeta\phi \in \DD$ and $\du{u_1}{\phi} = \du{u}{\zeta \phi}$ is finite.
  For continuity, let $\phi_k \to \phi$ in $\Ss$.
  Thus also $\zeta \phi_k \to \zeta \phi$ in $\DD$,
  since $\supp \zeta \phi_k \subset \supp \zeta$ is compact, 
  and for all $m \in \N$,
  \begin{equation}\nt
  \begin{split}
    \sup_{\substack{x \in \supp \zeta \\ |\alpha| \le m}}
      \big|\big(\partial^\alpha (\zeta & \phi_k) 
           - \partial^\alpha (\zeta \phi)\big)(x)|
    \\&
    \le \sup_{\substack{x \in \supp \zeta \\ |\alpha| \le m}}
      \sum_{\beta \le \alpha} c_{\alpha, \beta} \, 
      | \partial^\beta \zeta (x) | \, 
      | \partial^{\alpha - \beta} (\phi_k - \phi) (x) |
    \xrightarrow{k\to\infty} 0,
  \end{split}
  \ent\end{equation}
  because 
  $|\partial^{\alpha - \beta} (\phi_k - \phi)(x)| 
  \le \norm[0, \alpha - \beta]{\phi_k - \phi} \to 0$.
  Therefore $\du{u_1}{\phi_k - \phi} = \du{u}{\zeta\phi_k - \zeta\phi} \to 0$.

  Also $u_2$ is a tempered distribution because
  \begin{equation}\nt
    \nl{u_2}{1} \le \int_{|x| \ge 1} \frac{|K(x)|}{1 + |x|^2} \,dx
    = \int_{\Sm} |\Omega(\theta)| \,d\theta 
      \int_1^\infty \frac{dr}{r + r^3} < \infty.
  \ent\end{equation}
  Therefore $u = u_1 + u_2 \in \Ss'$. Now since
  \begin{equation}\nt
    \du{\pv K}{\phi} = \du{(1 + |x|^2) u}{\phi} = \du{u}{(1 + |x|^2) \phi}
  \ent\end{equation}
  and $(1 + |x|^2) \phi \in \Ss$ whenever $\phi \in \Ss$,
  we see that $\pv K$ is a tempered distribution.

  The estimate
  \begin{equation}
    |\du{\tilde{f} - K_{\eps, \infty} * f}{\phi}|
    \le \nl{\tilde{f} - K_{\eps, \infty} * f}{2} \nl{\phi}{2}
    \xrightarrow{\eps \to 0} 0,
  \end{equation}
  yields
  \begin{equation}
  \begin{split}
    \du{\tilde{f}}{\phi}
    &
    = \lim_{\eps \to \infty} \du{K_{\eps, \infty} * f}{\phi}
    \\&
    = \lim_{\eps \to \infty} \du{K_{\eps, \infty}}{\refl{f} * \phi}
    \\&
    = \lim_{\eps \to \infty} \int_{|y| > \eps} 
      K(y) \, \refl{f} * \phi(y) \,dy
    \\&
    = \du{\pv K}{\refl{f} * \phi}
    \\&
    = \du{\pv K * f}{\phi}.
  \end{split}
  \end{equation}
  This shows that \equ{LTwoLimEqualspvKConvf} holds.

  For proving \equ{FpvKConvfEqualsFpFpvKFf}, choose a sequence of functions
  $\zeta_k \in \Ss$ such that $\zeta_k \to \pv K$ in $\Ss'$. 
  Using the definition of convergence in $\Ss'$ and
  the convolution theorem \equ{ConvolutionTheorem} for two functions
  both in $\DD \subset L^1$, we get that for all $\phi \in \Ss$,
  \begin{equation}\nt
  \begin{split}
    \du{\F((\pv K) * f)}{\phi}
  & = \du{\pv K*f}{\widehat{\phi}}
    = \du{\pv K}{\refl{f} * \widehat{\phi}}
    = \lim_{k\to\infty} \du{\zeta_k}{\refl{f} * \widehat{\phi}}
  \\ & 
%    = \lim_{k\to\infty} \du{\zeta_k * f}{\widehat{\phi}}
    = \lim_{k\to\infty} \du{\F(\zeta_k * f)}{\phi}
    = \lim_{k\to\infty} \du{\Fp \widehat{\zeta_k} \Ff}{\phi}
  \\ &
    = \lim_{k\to\infty} \du{\Fp \zeta_k}{\F(\Ff \phi)}
    = \du{\Fp\, \pv K}{\F(\Ff \phi)}
  \\ &
    = \du{\Fp\, \F(\pv K)\, \F(f)}{\phi},
  \end{split}
  \ent\end{equation}
  as claimed.

  Aiming at establishing the fact that $\F(\pv K) = \FK$,
  construct a family of functions $g_\rho \in \DD$, $\rho > 0$,
  such that given any $R > 0$,
  \begin{equation}
    \widehat{g_\rho}(\xi) \neq 0
    \text{ for all } \xi \in B(0, R)
  \end{equation}
  when $\rho$ is sufficiently large.
  Such functions are given, for example, by
  \begin{equation}\nt
    g_\rho(x) 
    := \sigma_\rho \, \chim_{0,1}(x)
    =
    \begin{cases}
      \exp\left({1 + \frac{1}{\rho^2 |x|^2 - 1}}\right), & |x| < 1/\rho \\
      0,                                              & |x| \ge 1/\rho,
    \end{cases}
  \ent\end{equation}
%  where 
%  $\chim_{0,1}$ is the approximate characteristic function of
%  \equ{ApproxCharacteristicFunction} and 
%  $\rho > 0$ is sufficiently large:
  which can be seen as follows:
  Because 
  \begin{equation}\nt
    \widehat{g_1}(0) = \int g_1(x) \,dx > 0,
  \ent\end{equation}
  and $\widehat{g_1} \in \Ss$ is continuous,
  $\widehat{g_1}(\xi) \neq 0$ in some neighbourhood $B(0,\tilde{\rho})$.
  By \equ{FourierTransformOfDilation},
  $\widehat{g_\rho}(\xi) = \rho^{-n} \widehat{g_1}(\xi/\rho) \neq 0$ when
  $|\xi| < \rho \tilde{\rho}$. Hence, choosing $\rho \ge R/\tilde{\rho}$
  ensures that $\widehat{g_\rho}(\xi) \neq 0$ when $\xi \in B(0,R)$.

  Now for any $\phi \in \DD$, 
  let $R_\phi$ be such that $\supp \phi \subset B(0,R_\phi)$
  and choose $\rho$ so large that $\widehat{g_\rho}(\xi) \neq 0$ 
  for all $\xi \in B(0, R_\phi + 1)$.
  Then by \equ{LTwoLimEqualspvKConvf} and \equ{FpvKConvfEqualsFpFpvKFf},
  \begin{equation}
  \begin{split}
    \du{\F(\pv K)}{\phi}
    &
    = \Du{\F(\pv K)}{\frac{\widehat{g_\rho} \,\chim_{R_\phi, R_\phi + 1}}{
        \widehat{g_\rho}} \, \phi}
    \\&
    = \Du{\F(\pv K) \, \widehat{g_\rho}}{\frac{\chim_{R_\phi, R_\phi + 1}}{
        \widehat{g_\rho}} \, \phi}
    \\&
    = \Du{\Fm \, \F\Big(\ltwolim_{\substack{\eps \to 0\\ \eta \to \infty}} 
        K_{\eps, \eta} * g_\rho \Big)}{\frac{\chim_{R_\phi, R_\phi + 1}}{
        \widehat{g_\rho}} \, \phi},
  \end{split}
  \end{equation}
  whence Theorem~\ref{thm:CalderonZygmund}(c) implies that
  \begin{equation}
%  \begin{split}
    \du{\F(\pv K)}{\phi}
%    &
    = \Du{\FK \widehat{g_\rho}}{\frac{\chim_{R_\phi, R_\phi + 1}}{
        \widehat{g_\rho}} \, \phi}
%    \\&
    = \du{\FK}{\phi}.
%  \end{split}
  \end{equation}
  Consequently $\F(\pv K) = \FK$, which is essentially bounded by
  Theorem~\ref{thm:CalderonZygmund}(c).
\end{tod}


The principal results of the previous theorems can be summarized as
\begin{corollary}
  \label{cor:CalderonZygmundSummary}
  Let $n \in \Z_+$ and let $K:\Rn \to \C$ be of the form
  \begin{equation}\nt
    K(x) = \frac{\Omega(\frac{x}{|x|})}{|x|^n}
  \ent\end{equation}
  where $\Omega: \Sm \to \C$ is an odd, bounded function.
  Then the essentially bounded function 
  $\F(\pv K)$ is given almost everywhere by 
  \begin{equation}
    \label{CZSummaryFK}
    \F(\pv K)(\xi) = - i \pi\, \Fm\, \int_\spxi \Omega(\theta) \, d\theta.
  \end{equation}
  For all $f \in L^2$,
  \begin{equation}\nt
    \label{CZSummaryConvolutionTheorem}
    \F(\pv K * f) = \Fp \, \F(\pv K) \, \Ff
  \ent\end{equation}
  and
  \begin{equation}
    \label{CZSummaryBoundedness}
    \nl{\pv K * f}{2} \le \Fp \, \nl{\F(\pv K)}{\infty} \, \nl{f}{2}.
  \end{equation}
\end{corollary}



\section{Riesz Transforms}
\label{sec:RieszTransforms}

The results of the Calderón-Zygmund theory, derived in the previous section,
can now be applied to the Riesz transforms. This will enable us
to to calculate the derivatives of the Riesz potential,
and to finally show that $\Lambda \, R_1 * f = f$ 
in Chapter~\ref{chap:ReconstructionFormulae}.

\begin{theorem}
  \label{thm:FourierTransformOfRieszTransform}
  Let $n \in \Z_+$.
  For almost all $\xi \in \Rn$,
  \begin{equation}\nt
    \F\left(\pv \dodo{R_1}{x_j}\right)(\xi) = i \, \Fm \, \frac{\xi_j}{|\xi|}.
  \ent\end{equation}
  The Riesz transform $\pv \dodo{R_1}{x_j} *$ is a bounded operator 
  from $L^2(\Rn)$ to $L^2(\Rn)$, and
  \begin{equation}
    \label{FourierTransformOfRieszTransformConvolutionTheorem}
    \F\left(\pv \dodo{R_1}{x_j} * f\right)(\xi)
    = i \, \frac{\xi_j}{|\xi|} \, \Ff(\xi)
  \end{equation}
  for all $f \in L^2(\Rn)$.
\end{theorem}

\begin{tod}
  For $n \ge 2$, recall that
  \begin{equation}
    \dodo{R_1}{x_j}(x) 
    = b_n (1 - n) \, \frac{x_j}{|x|^{n+1}} 
    = b_n (1 - n) \, \frac{\theta_j}{|x|^n}.
  \end{equation}
  First assume that $|\xi| = 1$ and choose an orthonormal basis
  $(u^1, \ldots, u^n)$ of $\Rn$ with $u^1 = \xi$, so that
  \begin{equation}\nt
    \theta_j = \sum_{k=1}^n (\theta\cdot u^k)u^k_j 
    = (\theta\cdot\xi)\xi_j + \sum_{k = 2}^n (\theta\cdot u^k) u^k_j.
  \ent\end{equation}
  Using Corollary~\ref{cor:CalderonZygmundSummary}, 
  with $\Omega(\theta) = \theta_j$, 
  which is clearly an odd and bounded function, we have that almost everywhere
  \begin{multline}\nt
    \F\left(\pv \dodo{R_1}{x_j}\right)(\xi) 
    \\
    =
      - \frac{i \pi \, b_n \, (1-n)}{\Fp}
      \left( \xi_j \int_\spxi \theta\cdot\xi \,d\theta +
      \sum_{k = 2}^n u^k_j \int_\spxi \theta\cdot u^k \, d\theta \right).
  \ent\end{multline}
  For $k \neq 1$, the integrand $\theta\cdot u^k$ is positive on one
  half of $\spxi$ and negative but equal in absolute value on the other half,
  whence all but the first term in the sum vanish. 
  (See Figure~\ref{fig:fdr1dxj}.)

  \kuvienpoisto{
    \begin{figure}[htbp]
      \begin{center}
        \vspace{2ex}
        \leavevmode
        \epsfig{file=pics/fdr1dxj.eps, width=10cm}
        \caption{The integrand $\theta\cdot u^k$, $k \neq 1$, 
          is positive on one half of $\spxi$ 
          and negative but equal in absolute value on the other half.}
        \label{fig:fdr1dxj}
      \end{center}
    \end{figure}
    }

  Using the spherical coordinates of Lemma~\ref{lemma:SphericalCoordinatesSm}
  with $\theta\cdot\xi = x_1 = \cos\varphi$,
  \begin{equation}\nt
    \int_\spxi \theta\cdot\xi \,d\theta
    = |S^{n-2}|\, \int_0^{\pi/2} \cos\varphi \sin^{n-2} \varphi \,d\varphi
    = |S^{n-2}|\, \sij{0}{\pi/2} \frac{\sin^{n-1} \varphi}{n-1}
    = \frac{|S^{n-2}|}{n-1}.
  \ent\end{equation}
  Hence
  \begin{equation}\nt
    \F\left(\pv \dodo{R_1}{x_j}\right)(\xi) 
    = - \frac{i \pi}{\Fp} \, \frac{1}{\pi\,|S^{n-2}|}\,(1-n) \, \xi_j \,
      \frac{|S^{n-2}|}{n-1}
    = i \,\Fm\, \xi_j
  \ent\end{equation}
  as claimed, since we assumed that $|\xi| = 1$. If then $|\xi| \ne 1$
  and $\xi \neq 0$,
  we see from ~\equ{CZSummaryFK} that since $\spxi$ only depends on the
  direction of $\xi$ and not on its absolute value,
  \begin{equation}\nt
    \F\left(\pv \dodo{R_1}{x_j}\right)(\xi)
    = \F\left(\pv \dodo{R_1}{x_j}\right) \left(\frac{\xi}{|\xi|}\right)
    = i \,\Fm\, \left( \frac{\xi}{|\xi|} \right)_{\!\! j}
    = i \,\Fm\, \frac{\xi_j}{|\xi|}.
  \ent\end{equation}

  In the special case $n = 1$
  \begin{equation}
    \dodo{R_1}{x}(x) = \frac{\Omega(\frac{x}{|x|})}{|x|}
  \end{equation}
  where $\Omega(\theta) = -\theta/\pi$, and
  \begin{equation}
    S^0_+(\xi) 
    = \big\{\tfrac{\xi}{|\xi|}\big\} 
    =
    \begin{cases}
     \{ 1\}, & \xi > 0 \\
     \{-1\}, & \xi < 0.
    \end{cases}
  \end{equation}
  Therefore
  \begin{equation}
    \F\left(\pv \dodo{R_1}{x} \right)(\xi)
    = - \frac{i \pi}{\sqrt{2\pi}} 
      \int_{\{\frac{\xi}{|\xi}\}} - \frac{\theta}{\pi} \,d\theta
    = \frac{i}{\sqrt{2\pi}} \frac{\xi}{|\xi|}
  \end{equation}
  by Corollary~\ref{cor:CalderonZygmundSummary}, as claimed.

  Boundedness and \equ{FourierTransformOfRieszTransformConvolutionTheorem} 
  follow from \equ{CZSummaryBoundedness} and \equ{CZSummaryConvolutionTheorem}.
\end{tod}


We are now ready to derive the representation of $\Lambda f$ as
a sum of Riesz transforms of the derivatives of $f$.

\begin{theorem}
  \label{thm:LambdaAspvConvolution}
  For all $n \in \Z_+$, the Calderón operator $\Lambda$ can be expressed as
  \begin{equation}
    \label{LambdaAsPVConvolution}
    \Lambda = - \sum_{j=1}^n \pv \dodo{R_1}{x_j} * D_j.
  \end{equation}
  It is a continuous operator from $H^1(\Rn)$ to $L^2(\Rn)$.
\end{theorem}
\begin{tod}
  Write $\Lambda_0$ for the operator on the right hand side of 
  \equ{LambdaAsPVConvolution}.
  If $f \in H^1$, its first order distribution derivatives
  $D_j f \in L^2 \subset \Ss'$ satisfy
  \begin{equation}\nt
    \nl{D_j f}{2} = \nl{\F(D_j f)}{2} = \nl{\xi_j\, \Ff}{2}
    \le \nl{(1 + |\xi|^2)^{1/2}\, \Ff}{2} = \norm[H^1]{f}.
  \ent\end{equation}
  By Theorem~\ref{thm:CalderonZygmund},  
  $\nl{\pv \dodo{R_1}{x_j} * D_j f}{2} \le M_j\, \nl{D_j f}{2}$
  for some $M_j < \infty$. Therefore, 
  \begin{equation}\nt
    \nl{\Lambda_0 f}{2} \le C \norm[H^1]{f},
    \qquad \quad
    C = \sum_{j=1}^n M_j,
  \ent\end{equation}
  showing that $\Lambda_0: H^1 \to L^2$ is continuous.
  By Corollary~\ref{cor:CalderonZygmundSummary}, 
  Theorem~\ref{thm:FourierTransformOfRieszTransform} 
  and Equation~\equ{FourierTransformOfDerivatives},
  \begin{equation}
  \begin{split}
    \F \left( \pv \dodo{R_1}{x_j} * D_j f \right)
  & = \Fp\, \F\left(\pv \dodo{R_1}{x_j} \right) \F( D_j f)
    \\&
    = i\,\frac{\xi_j}{|\xi|}\, i\, \xi_j \,\Ff
    = - \frac{\xi_j^2}{|\xi|}\,\Ff
  \end{split}
  \end{equation}
  and hence
  \begin{equation}\nt
    \F (\Lambda_0 f) 
    = \frac{\sum_{j=1}^n \xi_j^2}{|\xi|}\,\Ff
    = |\xi|\,\Ff.
  \ent\end{equation}
  Therefore, $\Lambda_0 = \Lambda$.
\end{tod}


The following theorem shows that in the partial derivatives of the
Riesz potential, the derivative can be taken under the integral sign
if the integral is understood as a principal value integral.

Its proof uses Theorems \ref{thm:ZiemerThm284} and \ref{thm:ZiemerThm291},
for the proofs of which we only refer to 
\cite{stein70:sing} and \cite{ziemer89:weakly_diff}.
This is the only place in this work where a proof relies on 
outside references, save of course some very basic results.

\begin{theorem}
  \label{thm:DerivativeOfRieszPotential}
  If $f \in L^2(\Rn)$, then
  \begin{equation}
    D_j (R_1 * f) = \pv \dodo{R_1}{x_j} * f.
  \end{equation}
\end{theorem}

\begin{tod}
  Choose a sequence of test functions $\phi_k \in \DD$ with
  $\nl{f - \phi_k}{2} \xrightarrow{k \to \infty} 0$.
  Theorems \ref{thm:FourierTransformOfR1} and 
  \ref{thm:FourierTransformOfRieszTransform} assert that
  for all test functions $\psi \in \DD$, 
  \begin{equation}
  \begin{split}
    \label{DerivativeOfRieszPotential6}
    \du{D_j (R_1 * \phi_k)}{\psi}
    &
    = \Du{R_1 * \phi_k}{- \dodo{\psi}{x_j}}
    \\&
    = \Du{\Fp \, \F R_1(\xi) \widehat{\phi_k}(\xi)}{- 
                                           \Fi\left(\dodo{\psi}{x_j}\right)}
    \\&
    = \Du{|\xi|^{-1} \widehat{\phi_k}(\xi)}{i \, \xi_j \, \Fi(\psi)}
    \\&
    = \Du{i \, \frac{\xi_j}{|\xi|} \, \widehat{\phi_k}(\xi)}{\Fi(\psi)}
    \\&
    = \Du{\pv \dodo{R_1}{x_j} * \phi_k}{\psi},
  \end{split}
  \end{equation}
  so that $D_j (R_1 * \phi_k) = \pv \dodo{R_1}{x_j} * \phi_k$.
  By Theorem~\ref{thm:FourierTransformOfRieszTransform},
  \begin{equation}
    \nl{D_j (R_1 * \phi_k)}{2}
    = \Nl{\pv \dodo{R_1}{x_j} * \phi_k}{2}
    \le C_1 \, \nl{\phi_k}{2}
    \le C_1 \sup_{k \in \N} \nl{\phi_k}{2},
  \end{equation}
  which is finite since $\phi_k \to f$.
  As $L^2(\Rn)$ is a reflexive and separable Banach space,
  the Banach-Alaoglu theorem (Theorem~\ref{thm:BanachAlaoglu}) 
  implies the existence of a weakly convergent subsequence 
  $\left(D_j (R_1 * \phi_{k_l})\right)_{l\in\N}$.
  This means that there is a $v_j \in L^2(\Rn)$ such that
  for all test functions $\psi \in \DD$,
  \begin{equation}
    \label{DerivativeOfRieszPotentialWeakLimit}
    \du{D_j (R_1 * \phi_{k_l})}{\psi}
    = \ip{D_j (R_1 * \phi_{k_l})}{\conj{\psi}}
    \xrightarrow{l \to \infty}
    \ip{v_j}{\conj{\psi}}
    = \du{v_j}{\psi}.
  \end{equation}

  Corollary~\ref{cor:ZiemerCorollary} shows that 
  for all compact sets $E \subset \Rn$ there are constants 
  $q \ge 2$ and $C_2 > 0$ such that
  \begin{equation}
    \nl{\chi_E \, R_1 * g}{q} \le C_2 \, \nl{g}{2}
  \end{equation}
  for all $g \in L^2$, so that 
  $R_1 * f$ and $R_1 * \phi_{k}$ are in $\Lloc[q] \subset \Lloc \subset \Ss'$
  and
  \begin{equation}
    \Nl{\chi_E \, (R_1 * f - R_1 * \phi_k)}{q} 
    \le C_2 \, \nl{f - \phi_k}{2}
    \xrightarrow{k \to \infty} 0.
  \end{equation}
  Thus for all test functions $\psi \in \DD$,
  \begin{align}
    \left|\Du{R_1 * \phi_{k_l}}{\dodo{\psi}{x_j}} 
    - \Du{R_1 * f}{\dodo{\psi}{x_j}} \right|
    &
    \le \left| \int_{\supp \psi} \left(R_1 * \phi_{k_l} - R_1 * f\right) \,
      \dodo{\psi}{x_j} \, dx \right|
    \notag
    \\&
    \le \nl{\chi_{\supp \psi} \, (R_1 * \phi_{k_l} - R_1 * f)}{q} \,
        \nl{\dodo{\psi}{x_j}}{q'}
    \notag
    \\&
        \xrightarrow{l \to \infty} 0,
  \end{align}
  which shows that
  \begin{equation}
    \du{D_j (R_1 * \phi_{k_l})}{\psi}
    = \Du{R_1 * \phi_{k_l}}{- \dodo{\psi}{x_j}}
    \xrightarrow{l \to \infty} \Du{R_1 * f}{- \dodo{\psi}{x_j}}
    = \Du{D_j (R_1 * f)}{\psi}.
  \end{equation}
  When combined with \equ{DerivativeOfRieszPotentialWeakLimit}, 
  this implies that 
  \begin{equation}
    \du{v_j}{\psi} = \du{D_j (R_1 * f)}{\psi}
  \end{equation}
  for all $\psi$ in the dense set $\DD \subset L^2$,
  which implies that $D_j (R_1 * f) \in L^2$
  is the weak limit of the subsequence 
  $\left(D_j (R_1 * \phi_{k_l})\right)_{l \in \N}$.

  Again by Theorem~\ref{thm:FourierTransformOfRieszTransform},
  \begin{equation}
    \Nl{\pv \dodo{R_1}{x_j} * \phi_{k_l} - \pv \dodo{R_1}{x_j} * f}{2}
    \le C_1 \, \nl{\phi_{k_l} - f}{2}
    \xrightarrow{k \to \infty} 0,
  \end{equation}
  so that
  \begin{equation}
    D_j (R_1 * \phi_{k_l})
    = \pv \dodo{R_1}{x_j} * \phi_{k_l}
    \xrightarrow{l \to \infty} \pv \dodo{R_1}{x_j} * f
    \quad
    \text{strongly.}
  \end{equation}
%  the sequence of functions 
%  $D_j (R_1 * \phi_{k_l}) = \pv \dodo{R_1}{x_j} * \phi_{k_l}$ has the
%  strong limit $\pv \dodo{R_1}{x_j} * f$. 
  Because strong convergence implies weak convergence,
  we conclude that $D_j (R_1 * f) = \pv \dodo{R_1}{x_j} * f$.
\end{tod}




\chapter{Derivation of the Reconstruction Formulae}
\label{chap:ReconstructionFormulae}


We are now ready to derive rigorously the reconstruction formulae
of Theorems \ref{thm:ApproxParBeamReconstructionChap3}
and \ref{thm:EConvLambdaFReconstructionChap3},
which will be repeated later on 
as Corollary~\ref{cor:ApproxReconstruction}
and Theorem~\ref{thm:EConvLambdaFReconstruction}, respectively.
We shall follow the treatment in \cite{smith85:mathfound}.

\section{The X-ray Domain $\Dxr$}
\label{sec:XRayDomain}

The function to be reconstructed will be allowed to be in
the space $\Dxr$, defined as follows:

\begin{definition}
  \label{def:Dxr}
  For $n \ge 2$, the space
  \begin{equation}
    \Dxr(\Rn) = \{ g \in L^2(\Rn) \,|\, (1 + |x|)^{1-n} g \in L^1(\Rn) \}
  \end{equation}
  with norm
  \begin{equation}
    \nxr{g} = \nl{g}{2} + \nl{(1 + |x|)^{1-n} g(x)}{1}
  \end{equation}
  is called the \emph{x-ray domain} over $\Rn$.
\end{definition}

This is a useful domain because $R_1 * f$ behaves nicely for such $f$
that $(1 + |x|)^{1-n} f \in L^1$, as will be shown shortly in 
Theorem~\ref{thm:SKthm38}.
The requirement that the function belong to $L^2$ is natural
since the derivations of the properties of
the Riesz transform $\dodo{R_1}{x_j} *$ and the Calderón operator $\Lambda$
were done in the previous chapter for $L^2$ spaces.

Note that when $n > 2$ and $f \in L^2(\Rn)$,
\begin{equation}
\begin{split}
  \nl{(1 + |x|)^{1-n} f(x)}{1}^2
  &
  \le \nl{(1 + |x|)^{1-n}}{2}^2 \nl{f}{2}^2
  \\&
  = |\Sm| \int_0^\infty \frac{r^{n-1}}{(1 + r)^{2n-2}} \,dr \, \nl{f}{2}^2
  \\&
  \le |\Sm| \left[ \int_0^1 \,dr
    + \int_1^\infty \frac{r^{n-1}}{2^{2n-2} r^{2n-2}} \,dr \right] \nl{f}{2}^2
  < \infty
\end{split}
\end{equation}
so that $\Dxr$ is simply $L^2$. 
In fact after proving Theorem~\ref{thm:SKthm38} 
we will be able to show using Corollary~\ref{cor:ZiemerCorollary}
that this is also true when $n = 2$.
However, since Corollary~\ref{cor:ZiemerCorollary} uses
Theorems \ref{thm:ZiemerThm284} and \ref{thm:ZiemerThm291}
whose proofs are not presented in this work, 
we shall continue to speak of $\Dxr$ 
following the treatment in \cite{smith85:mathfound},
in attempt to minimize the number of places where proofs rely on
outside references.
This is also expedient for emphasizing that the difference between
the norms $\nl{\cdot}{2}$ and $\nxr{\cdot}$.


In showing that $R_1 * f$ is well-behaved for $f \in \Dxr$,
we shall need the following result:

\begin{lemma}
  \label{lemma:SKlemma36}
  If $n \ge 2$, 
  $\phi \in L^1(\Rn)$ and $(1 + |x|)^n \phi(x) \in L^\infty(\Rn)$,
  then
  \begin{equation}
    |R_1 * \phi(x)| 
    \le C \, (1 + |x|)^{1-n} 
          \big[ \nl{\phi}{1} + \nl{(1 + |y|)^n \phi(y)}{\infty} \big]
  \end{equation}
  for some constant $C > 0$, independent of $\phi$.
\end{lemma}

\begin{tod}
  It suffices to prove the claim for $\psi(x) := |\phi(x)|$
  because then
  \begin{equation}
  \begin{split}
    |R_1 * \phi(x)| 
    &
    \le |R_1| * |\phi|(x)
    = R_1 * \psi(x)
    \\ &
    \le C \, (1 + |x|)^{1-n} 
          \left( \nl{\psi}{1} + \nl{(1 + |y|)^n \psi(y)}{\infty} \right)
    \\ &
    = C (1 + |x|)^{1-n} 
          \left( \nl{\phi}{1} + \nl{(1 + |y|)^n \phi(y)}{\infty} \right).
  \end{split}
  \end{equation}

  Observe that
  \begin{equation}
    \psi(x) = (1 + |x|)^{-n}\, (1 + |x|)^n \, \psi(x)
    \le M\, (1 + |x|)^{-n},
  \end{equation}
  where $M := \nl{(1 + |y|)^n\psi(y)}{\infty}$ is finite by assumption.
  Distinguish two cases, $|x| < 1$ and $|x| \ge 1$.

  When $|x| < 1$, divide the range of integration into two parts:
  \begin{equation}
    R_1 * \psi(x) 
    = b_n \int_{|x-y| \le 1} \frac{\psi(y)}{|x-y|^{n-1}} \,dy
      + b_n \int_{|x-y| > 1} \frac{\psi(y)}{|x-y|^{n-1}} \,dy.
  \end{equation}
  In the first integral, make the change of variable $z = y - x$, $dz = dy$
  and note that $\psi(y) \le M (1 + |x|)^{-n} \le M$.
  In the second integral, $1 / |x - y|^{n-1} \le 1$. Therefore,
  \begin{equation}
  \begin{split}
    R_1 * \psi(x)
    &
    \le b_n M \int_{|z| \le 1} \frac{dz}{|z|^{n-1}} 
        + b_n \int_{|x-y| > 1} \psi(y) \,dy
    \\ &
    \le b_n M \,|\Sm| \, \int_0^1 r^{1-n} r^{n-1} \,dr
        + b_n \nl{\psi}{1}
    \\ &
    \le b_n \, |\Sm| \, \big(M + \nl{\psi}{1}\big)   
    \\ &
    \le 2^{n-1} b_n \, |\Sm| \, (1 + |x|)^{1-n} \big(M + \nl{\psi}{1}\big),
  \end{split}
  \end{equation}
  as $2^{n-1} \, (1 + |x|)^{1-n} \ge 1$ for $|x| < 1$.

  For $|x| \ge 1$, divide the domain of integration into three parts:
  \begin{equation}
    \label{SKlemma36part2}
    R_1 * \psi(x) 
    = \int_{|y| \le \frac{|x|}{2}} \frac{b_n \, \psi(y) \,dy}{|x-y|^{n-1}}
      + \int_{\frac{|x|}{2} < |y| < 2\,|x|} 
             \frac{b_n \, \psi(y) \,dy}{|x-y|^{n-1}}
      + \int_{|y| \ge 2\,|x|} \frac{b_n \, \psi(y) \,dy}{|x-y|^{n-1}}.
  \end{equation}

  When $|y| \le |x|/2$, 
  \begin{equation}
    |x - y| \ge ||x| - |y|| = |x| - |y| \ge |x| - \frac{|x|}{2} = \frac{|x|}{2}
  \end{equation}
  and the first integral is majorized by
  \begin{equation}
    \label{SKlemma36part2a}
    \frac{2^{n-1}\, b_n}{|x|^{n-1}} 
    \int_{|y| \le \frac{|x|}{2}} \psi(y) \,dy
    \le \frac{2^{n-1}\, b_n}{|x|^{n-1}} \, \nl{\psi}{1}.
  \end{equation}

  When $|x|/2 < |y| < 2\,|x|$, write $x = |x|\,\theta$, $\theta \in \Sm$,
  and make the change of variable $y = |x| \,z$, $dy = |x|^n \, dz$, with 
  $|z| = |y|/|x| \in (\frac{1}{2}, 2)$.
  Then $|x - y| = |x| |\theta - z|$ and the integral becomes
  \begin{equation}
    \frac{b_n}{|x|^{n-1}}
    \int_{\frac{1}{2} < |z| < 2} 
             \frac{\psi(|x| \, z)\, |x|^n}{|\theta-z|^{n-1}} \,dz.
  \end{equation}
  The estimates $|x|^n = (|x||z|)^n \, |z|^{-n} \le (1 + |x||z|)^n \,2^n$
  and 
  \begin{equation}
    \psi(|x| \,z) = (1 + |x||z|)^{-n} \, (1 + |x||z|)^n \,\psi(|x| \,z)
    \le M\, (1 + |x||z|)^{-n} 
  \end{equation}
  show that the second integral is less than or equal to
  \begin{equation}
  \begin{split}
    \label{SKlemma36part2b}
%    \frac{b_n}{|x|^{n-1}}
%    &
%    \int_{\frac{1}{2} < |z| < 2} 
%      \frac{M\, (1 + |x||z|)^{-n} (1 + |x||z|)^n \,2^n}{|\theta-z|^{n-1}} \,dz
%    \\&
%    \le
    \frac{2^n\, M\, b_n}{|x|^{n-1}} 
      \int_{\frac{1}{2} < |z| < 2} \frac{dz}{|\theta - z|^{n-1}}
    &
    \le \frac{2^n\, M\, b_n}{|x|^{n-1}} \int_{|w| < 3} \frac{dw}{|w|^{n-1}}
    \\&
    = \frac{2^n\, M\, b_n}{|x|^{n-1}} \,|\Sm|\,
      \int_0^3 \frac{r^{n-1} \,dr}{r^{n-1}}
    \\&
    = \frac{2^n\, 3 \, M\, b_n \, |\Sm|}{|x|^{n-1}}.
  \end{split}
  \end{equation}

  Finally, when $|y| \ge 2\,|x|$, the situation is much like when
  $|y| \le |x|/2$: Now
  $|x - y| \ge ||x| - |y|| = |y| - |x| \ge 2\,|x| - |x| = |x|$,
  and the third integral has
  \begin{equation}
    \label{SKlemma36part2c}
    \frac{b_n}{|x|^{n-1}} \int_{|y| \ge 2\, |x|} \psi(y) \,dy
    \le \frac{b_n}{|x|^{n-1}} \, \nl{\psi}{1}
  \end{equation}
  as an upper bound.

  The upper bounds \equ{SKlemma36part2a}, \equ{SKlemma36part2b} and
  \equ{SKlemma36part2c} for the three integrals in \equ{SKlemma36part2}
  combine to show that the claim holds:
  \begin{equation}
    \begin{split}
    R_1 * \psi(x)
    &
    \le \frac{(2^{n-1} + 1)\, b_n}{|x|^{n-1}} \,\nl{\psi}{1}
      + \frac{2^n \, 3 \, b_n \,|\Sm|}{|x|^{n-1}} \, M
    \\&
    \le 2^n \, 3 \, b_n \,|\Sm| \, (1 + |x|)^{n-1} \,
        \big(\nl{\psi}{1} + M \big).
  \end{split}
  \end{equation}
\end{tod}


This result helps to show that the condition that $(1 + |x|)^{1-n} f \in L^1$
is both necessary and sufficient for $R_1 * f$ to converge absolutely
almost everywhere:

\begin{theorem}
  \label{thm:SKthm38}
  If $n \ge 2$ and $(1 + |x|)^{1-n} f(x) \in L^1(\Rn)$, 
  then the convolution integral
  $R_1 * f (x)$ converges absolutely for almost almost all $x \in \Rn$,
  and $R_1 * f \in \Lloc(\Rn)$. Furthermore, there is a constant $C > 0$
  independent of $f$, such that
  \begin{equation}
    \label{SKthm38estimate}
%    |\du{R_1 * f}{\phi}| 
    \Big| \int_\Rn R_1 * f(x) \, \phi(x) \,dx \, \Big|
    \le \Nl{(1 + |x|)^{1-n} f}{1} 
        \big[ \nl{\phi}{1} + \nl{(1 + |x|)^n \phi}{\infty} \big]
  \end{equation}
  for all measurable functions $\phi: \Rn \to \C$.
  Conversely, if the convolution integral $R_1 * f(x_0)$ converges absolutely
  for some $x_0 \in \Rn$, then $(1 + |x|)^{1-n} f \in L^1(\Rn).$
\end{theorem}

\begin{tod}
  The triangle inequality and Fubini's theorem for non-negative functions
  show that
  \begin{equation}
  \begin{split}
    \label{SKthm38DuInSpe}
    \Big| \int_\Rn R_1 * f(x) \, \phi(x) \,dx \,\Big|
    &
    = \Big| \int_\Rn \int_\Rn R_1(x-y) f(y) \phi(x) \,dy \,dx \,\Big|
    \\&
    \le \int_\Rn \int_\Rn R_1(x-y) |f(y)| |\phi(x)| \,dy \,dx
    \\&
    = \int_\Rn |f(y)| \, \int_\Rn R_1(y-x) \, |\phi(x)| \,dx \, dy.
  \end{split}
  \end{equation}
  By right of Lemma~\ref{lemma:SKlemma36},
%  \begin{equation}
%  \begin{split}
  \begin{multline}
    \int_\Rn |f(y)| \, \int_\Rn R_1(y-x) \, |\phi(x)| \,dx \, dy
%    \\&
    \\
    \le C \int_\Rn |f(y)| \, (1 + |y|)^{1-n} \, dy \,
        \big[\nl{\phi}{1} + \nl{(1 + |y|)^n \phi(y)}{\infty} \big],
%    \\&
%    \le C \, \Nl{(1 + |y|)^{1-n} f(y)}{1}
%        \big[\nl{\phi}{1} + \nl{(1 + |y|)^n \phi(y)}{\infty} \big],
%  \end{split}
%  \end{equation}
  \end{multline}
  which proves \equ{SKthm38estimate}
  and also shows that $R_1 * f \in \Lloc$ indeed: 
  For any compact $K \subset \Rn$,
  \begin{equation}
  \begin{split}
    \label{SKthm38LocalIntegrability}
    \int_K |R_1 * f (x)| \,dx
    &
    \le \int_\Rn R_1 * |f|(x) \, \chi_K(x) \,dx
    \\ &
    \le C \, \Nl{(1 + |y|)^{1-n} f}{1}
      \big[\nl{\chi_K}{1} + \nl{(1 + |y|)^n \chi_K(y)}{\infty} \big] < \infty.
  \end{split}
  \end{equation}

  This also implies absolute convergence almost everywhere, for if
  there were a set $E \subset \Rn$ with $m(E) > 0$ such that
  $R_1 * |f|(x) = \infty$ for all $x \in E$, then
  the regularity of the Lebesgue measure would assure the existence
  of a compact set $K \subset E$ with $m(K) \ge m(E) - m(E)/2 > 0$,
  and consequently
  \begin{equation}
    \int_K R_1 * |f|(x) = \infty,
  \end{equation}
  in contradiction with the local integrability just proved.

  For the converse claim, note that if $|x_0| \le 1$,
  $1 + |y| \ge |x_0| + |y| \ge |x_0 - y|$, 
  and if $|x_0| > 1$, 
  \begin{equation}
    1 + |y| 
    \ge 1 + \frac{|y|}{|x_0|} 
    = \frac{|x_0| + |y|}{|x_0|} 
    \ge \frac{|x_0 - y|}{|x_0|}.
  \end{equation}
  Therefore, $1 + |y| \ge \min \{1, |x_0|^{-1}\} \, |x_0 - y|$ and
  \begin{equation}
    \int_\Rn \left| (1 + |y|)^{1-n} \, f(y) \right|\, dy
    \le \max \{1, |x_0|^{n-1} \} \int_\Rn \frac{|f(y)|}{|x_0 - y|^{n-1}} \,dy
  \end{equation}
  is finite if $R_1 * f (x_0)$ converges absolutely.
\end{tod}


It now follows that the x-ray domain $\Dxr$ is actually
just the space of square-integrable functions;
the requirement that $(1 + |x|)^{1-n} f(x) \in L^1$ is satisfied
by all functions $f \in L^2$:

\begin{corollary}
  For all $n \ge 2$, $\Dxr(\Rn) = L^2(\Rn)$.
\end{corollary}
\vspace{-4ex}
\begin{tod}
If $f \in L^2$, then also $|f| \in L^2$, so that 
Corollary~\ref{cor:ZiemerCorollary} asserts that 
$R_1 * |f| \in \Lloc[q] \subset \Lloc$. 
Therefore the convolution integral $R_1 * f$ must converge absolutely 
almost everywhere.
As this is equivalent to the condition that $(1 + |x|)^{1-n} f(x) \in L^1$
by Theorem~\ref{thm:SKthm38}, we see that $f \in \Dxr$.
The converse inclusion is clear from the definition of $\Dxr$.
\end{tod}

As already stated after Definition~\ref{def:Dxr} of the x-ray domain $\Dxr$,
we shall not use this result
because its proof refers through Corollary~\ref{cor:ZiemerCorollary} 
to Theorems \ref{thm:ZiemerThm284} and \ref{thm:ZiemerThm291}
whose proofs are not presented in this work.

Another property of the x-ray domain is 
its stability with respect to mollifying with certain kernels:

\begin{theorem}
  \label{thm:SKlemma311}
  If $n \ge 2$, $g \in \Dxr(\Rn)$ and $(1 + |x|)^{1-n} e(x) \in L^1(\Rn)$,
  then $e*g \in \Dxr(\Rn)$ and
  \begin{equation}
    \nxr{e*g} \le C \, \nl{(1 + |x|)^{1-n} e(x)}{1} \nxr{g}
  \end{equation}
  for some constant $C > 0$ independent of $g$ and $e$.
\end{theorem}

\begin{tod}
  Young's and Hölder's inequalities 
  (Theorems \ref{thm:Young} and \ref{thm:Holder}) assert that
  \begin{equation}
  \label{SKlemma311part1}
  \begin{split}
    \nl{e*g}{2} 
    &
    \le \nl{e}{1} \nl{g}{2}
    \\&
    = \nl{(1 + |x|)^{1-n} (1 + |x|)^{n-1} e}{1} \nl{g}{2}
    \\&
    \le \nl{(1 + |x|)^{1-n}}{\infty} \nl{(1 + |x|)^{n-1} e}{1} \nl{g}{2}
%    \\&
%    = \nl{(1 + |x|)^{n-1} e}{1} \nl{g}{2}
    \\&
    \le \nl{(1 + |x|)^{n-1} e}{1} \nxr{g}.
  \end{split}
  \end{equation}
  which settles the estimate for the $L^2$ part of
  $\nxr{e * g} = \nl{e * g}{2} + \nl{(1 + |x|)^{1-n}\, e * g}{1}$.

  For the other part, using the fact that
  \begin{equation}
  \begin{split}
    (1 + a^2)(1 + b)^2
    &
    = 1 + 2 b + b^2 + a^2 + 2ab + a^2 b^2
    \\&
    = 1 + (a + b)^2 + 2 b (1 - a + a^2) + a^2 b^2
    \\&
    \ge 1 + (a + b)^2
  \end{split}
  \end{equation}
  for $b \ge 0$, with $a = |x|$ and $b = |y|$, gives
  \begin{equation}
    1 + |x - y|^2 \le 1 + (|x| + |y|)^2
    \le \left(1 + |x|^2 \right) (1 + |y|)^2
  \end{equation}
  and consequently
  \begin{equation}
    \label{SKlemma311ElementaryEstimate}
    \left(1 + |x|^2\right)^{\frac{1-n}{2}} 
    \le \left(1 + |x - y|^2\right)^{\frac{1-n}{2}} (1 + |y|)^{n-1}.
  \end{equation}
  As $1 + |x| \le (1 + |x|^2)^{1/2}$,
  estimate \equ{SKlemma311ElementaryEstimate} and Young's inequality yield
  \begin{equation}
  \begin{split}
    \Nl{(1 + |x|)^{1-n} e*g}{1}
    &
    \le \bnl{\left(1 + |x|^2\right)^{\frac{1-n}{2}} e * g(x)}{1}
    \\&
    \le \int_\Rn \int_\Rn 
        |e(y)||g(x-y)| \, \left(1 + |x|^2\right)^{\frac{1-n}{2}} \,dy \,dx
    \\&
    \le \int_\Rn \int_\Rn
        \left(1 + |x-y|^2\right)^{\frac{1-n}{2}} |g(x-y)|
        \left(1 + |y|\right)^{n-1} |e(y)| \,dy \,dx
    \\&
    = \nl{e_1 * g_1}{1}
    \le \nl{e_1}{1} \nl{g_1}{1},
  \end{split}
  \end{equation}
  where
  \begin{equation}
    e_1(x) = \left(1 + |x|\right)^{n-1} \, |e(x)|
    \qquad \text{and} \qquad
    g_1(x) = \left(1 + |x|^2\right)^{\frac{1-n}{2}} \,|g(x)|.
  \end{equation}
  Since
  \begin{equation}
    \nl{g_1}{1}
    = \Nl{\left(\frac{1 + 2\,|x| + |x|^2}{1 + |x|^2}\right)^{\frac{n-1}{2}}
          (1 + |x|)^{1-n} g(x)}{1}
    \le 2^{\frac{n-1}{2}} \Nl{(1 + |x|)^{1-n} g(x)}{1},
  \end{equation}
  this shows that
  \begin{equation}
  \begin{split}
    \Nl{(1 + |x|)^{1-n} e*g}{1}
    &
    \le 2^{\frac{n-1}{2}} \Nl{(1 + |x|)^{n-1} e}{1} \Nl{(1 + |x|)^{1-n} g}{1}
    \\&
    \le 2^{\frac{n-1}{2}} \Nl{(1 + |x|)^{n-1} e}{1} \Nxr{g},
  \end{split}
  \end{equation}
  which together with \equ{SKlemma311part1} implies the claim
  with $C = 2^{(n-1)/2} + 1$.
%  \begin{equation}
%  \begin{split}
%    \nxr{e*g}
%    &
%    = \nl{e*g}{2} + \Nl{(1 + |x|)^{1-n} (e*g)}{1}
%    \\&
%    \le \Nl{(1 + |x|)^{n-1} e}{1} \nxr{g} + 
%        2^{\frac{n-1}{2}} \Nl{(1 + |x|)^{n-1} e}{1} \Nxr{g}
%    \\&
%    \le \left(2^{\frac{n-1}{2}} + 1\right) \Nl{(1 + |x|)^{n-1} e}{1} \Nxr{g}.
%  \end{split}
%  \end{equation}
\end{tod}


It is pleasant to know that elements of $\Dxr$ can be approximated
using smooth functions with bounded support.
%\kommentti{\cite{smith85:mathfound}:ssa vain $C_0$!}

\begin{lemma}
  \label{lemma:CInfty0IsDenseInDxr}
  If $n \ge 2$ and $f \in \Dxr(\Rn)$, then
  \begin{equation}
    \nxr{f - \chi_{B(0,R)} f} \to 0
  \end{equation}
  as $R \to \infty$. 
  Furthermore, $C^\infty_0(\Rn)$ is dense in $\Dxr(\Rn)$.
\end{lemma}

\begin{tod}
  Fix $f \in \Dxr$ and $\eps > 0$.
  As $(1 + |x|)^{1-n} f \in L^1$,
  \begin{equation}
    \int_{|x| > r_1} (1 + |x|)^{1-n} \, |f(x)| \,dx < \frac{\eps}{4}
  \end{equation}
  for some $r_1 > 0$. Analogously, as $f \in L^2$,
  \begin{equation}
    \int_{|x| > r_2} |f(x)|^2 \,dx < \frac{\eps^2}{16}
  \end{equation}
  for some $r_2 > 0$. Thus for $R \ge \max\{ r_1, r_2 \} =: r_0$,
  \begin{equation}
    \nxr{f - \chi_{B(0,R)} f}
    = \int_{|x| > R} (1 + |x|)^{1-n} \, |f(x)| \,dx
      + \left( \int_{|x| > R} |f(x)|^2 \,dx \right)^{\frac{1}{2}}
    < \frac{\eps}{2},
  \end{equation}
  which proves the first claim. 

  Then set $f_0 = \chi_{B(0, r_0)} f$ and $\phi_\rho = e_\rho * f_0$ 
  with $e_\rho$ the approximate delta function of \equ{def:ApproxDelta}, 
  $\rho > 0$.
  As described in Section~\sectref{sec:LpMollification}, 
  $\phi_\rho \in C^\infty$ and $\nl{f_0 - \phi_\rho}{p} \to 0$ as $\rho \to 0$,
  for all $1 \le p < \infty$ for which $f_0 \in L^p$.
  Because 
  $\nl{f_0}{1} \le \nl{\chi_{B(0, \rho)}}{2} \nl{f}{2} < \infty$,
  \begin{align}
    \Nl{(1 + |x|)^{1-n} \left(f_0 - \phi_\rho\right)}{1}
    \le \Nl{f_0 - \phi_\rho}{1} < \frac{\eps}{4}
    & \qquad \text{when } \rho \ge \rho_1
    \\ 
  \intertext{for some $\rho_1 > 0$, and because 
    $\nl{f_0}{2} \le \nl{f}{2} < \infty$,
  }
    \Nl{f_0 - \phi_\rho}{2} < \frac{\eps}{4}
    & \qquad \text{when } \rho \ge \rho_2
  \end{align}
  for some $\rho_2 > 0$. 
  Choosing $\phi = \phi_{\max \{\rho_1, \rho_2\}}$ yields
  \begin{equation}
    \nxr{f - \phi} 
    \le \nxr{f - f_0} + \nxr{f_0 - \phi}
    < \frac{\eps}{2} + \frac{\eps}{4} + \frac{\eps}{4}
    = \eps.
  \end{equation}
  Since
  \begin{equation}
    \supp \phi_\rho 
    \subset \left\{ x + y \in \Rn \,|\, 
                        x \in \supp e_\rho, \, y \in \supp f_0 \right\}
    \subset B(0, R + \rho)
  \end{equation}
  is compact, $\phi \in C_0^\infty$.
\end{tod}



Before showing that the relation $\F(R_1 * f) = |\xi|\, \Ff$,
formally derived in Chapter~\sectref{chap:XRayImaging},
holds in $\Dxr$, we still prove three auxiliary results.

The first one shows that the Riesz potential $R_1 * g$ of a non-negative 
x-ray attenuation coefficient $g$ has the superharmonic-type property
that its average over a ball is at most as large as 
a constant times its value at the centre of the ball. \cite{frostman}


\begin{lemma}[Frostman's mean value theorem]
  \label{lemma:Frostman}
  Let $n \ge 2$, $h = \chi_{B(0,1)}$ and
  \begin{equation}
    h_\rho(x) = \rho^{-n} h(\pfrac{x}{\rho}) =
    \begin{cases}
      \rho^{-n}, & |x| < \rho \\
      0,         & |x| \ge \rho.
    \end{cases}
  \end{equation}
  Then there is a constant $C > 0$, independent of $\rho$, 
  such that for all $f \in \Dxr(\Rn)$ and $\rho > 0$,
  \begin{equation}
    |\du{R_1 * f}{h_\rho}| \le C \, R_1 * |f| (0).
  \end{equation}
\end{lemma}

\begin{tod}
  Fubini's theorem shows that since the integrand is non-negative,
  \begin{equation}
  \begin{split}
    |\du{R_1 * f}{h_\rho}| 
    &
    \le b_n \, \rho^{-n} 
        \int_{B(0,\rho)} \int_\Rn \frac{|f(y)|}{|x - y|^{n-1}} \,dy \,dx
    \\&
    = b_n \int_\Rn \frac{|f(y)|}{|y|^{n-1}} \, I(y,\rho) \,dy,
  \end{split}
  \end{equation}
  where
  \begin{equation}
    I(y, \rho) 
    = \frac{|y|^{n-1}}{\rho^n} \int_{B(0,\rho)} \frac{dx}{|x-y|^{n-1}}.
  \end{equation}
  It suffices to show that $I(y, \rho)$ is bounded for all 
  $y \in \Rn$, $\rho > 0$.

  Distinguish two cases, according to whether $|y| < 2\rho$ or $|y| \ge 2\rho$.
  In the first case,
  \begin{equation}
  \begin{split}
    0 \le I(y,\rho)
    &
    \le \frac{|y|^{n-1}}{\rho^n} \int_{B(y, 3\rho)} \frac{dx}{|x - y|^{n-1}}
    \\&
    =   \frac{|y|^{n-1}}{\rho^n} \, |\Sm| 
        \int_0^{3\rho} \frac{r^{n-1} \, dr}{r^{n-1}}
    \\&
    = \frac{|y|^{n-1} \, 3\rho}{\rho^n} \, |\Sm|
    \\&
    \le 2^{n-1} 3 \,|\Sm|.
  \end{split}
  \end{equation}
  For the second case $|y| \ge 2\rho$, note that $x \in B(0,\rho)$ implies that
  $|x| \le \rho \le |y|/2$ and thus
  \begin{equation}
    |x - y| \ge \big| |y| - |x| \big| \ge |y| - \frac{|y|}{2} = \frac{|y|}{2}.
  \end{equation}
  Therefore,
  \begin{equation}
    0 \le I(y, \rho)
    \le \frac{|y|^{n-1}}{\rho^n} 
        \int_{B(0,\rho)} \frac{2^{n-1}}{|y|^{n-1}} \,dx
    = \frac{2^{n-1} \,m(B(0,\rho))}{\rho^n} = 2^{n-1} \, |B^n|.
  \end{equation}
  This shows that in both cases,
  \begin{equation}
    I(y, \rho) 
    \le \max \left\{ 2^{n-1} 3 \,|\Sm|, \, 2^{n-1} \, |B^n| \right\}
    =: C.
  \end{equation}
\end{tod}


The second auxiliary result is that the (weighted) averages of 
$f$ and $R_1 * f$ over a bounded set tend to zero 
as the diameter of the set grows.
This indicates that in some respect, the mass of those functions is not
concentrated at infinity.

\begin{lemma}
  \label{lemma:SKlemma314}
  For any $e \in L^\infty_0(\Rn)$, define 
  $e_\rho(x) = \rho^{-n} e(\frac{x}{\rho})$, $\rho > 0$.
  Then
  \vspace{-2ex}
  \begin{enumerate}
  \samepage
  \item[(a)]
    if $n \in \Z_+$ and $\Ff \in L^1(\Rn) + L^2(\Rn)$, 
    then $\du{f}{e_\rho} \to 0$ as $\rho \to \infty$
  \item[(b)]
    if $n \ge 2$ and $f \in \Dxr(\Rn)$, 
    then $\du{R_1*f}{e_\rho} \to 0$ as $\rho \to \infty$.
  \end{enumerate}
\end{lemma}

\begin{tod}
  For claim (a), write $f = f_1 + f_2$
  with $\widehat{f_1} \in L^1$ and $\widehat{f_2} \in L^2$.
  The claim for $f_2$ follows from the estimate
  \begin{equation}
  \begin{split}
    |\du{f_2}{e_\rho}|
    &
%    \le \du{|f_2|}{|e_\rho|}
%    = \nl{f_2 e_\rho}{1}
    \le \nl{f_2 e_\rho}{1}
    \le \nl{f_2}{2} \nl{e_\rho}{2}
    \\&
    = \nl{f_2}{2} \, \sqrt{\int_\Rn \rho^{-2n} \, |e(\pfrac{x}{\rho})|^2 \,dx}
    \\&
    = \nl{\widehat{f_2}}{2} \, \rho^{-n/2} \sqrt{\int_\Rn |e(y)|^2 \,dy}
    \\&
%    = \rho^{-n/2} \,\nl{\widehat{f_2}}{2} \, \sqrt{\nl{e^2}{1}}
    \le \rho^{-n/2} \,\nl{\widehat{f_2}}{2}\,\nl{e}{\infty} \,\sqrt{m(\supp e)}
    \xrightarrow{\rho \to \infty} 0,
  \end{split}
  \end{equation}
  where the change of variable $y = x/\rho$, $dy = \rho^{-n} \,dx$ 
  has been used.

  For estimating $\du{f_1}{e_\rho}$, use Fubini's theorem to see that
  \begin{equation}
  \begin{split}
    \du{f_1}{e_\rho} 
    &
    = \int \Fm \int e^{i x \cdot \xi} \widehat{f_1}(\xi) \,d\xi
               \, e_\rho(x) \,dx
    \\&
    = \Fm \int \int e^{i x \cdot \xi} \rho^{-n} e(\pfrac{x}{\rho}) \,dx
               \, \widehat{f_1}(\xi) \, d\xi
    \\&
    = \Fm \int \int e^{-i y \cdot (-\rho \xi)} e(y) \,dy
               \, \widehat{f_1}(\xi) \, d\xi
    \\&
    = \Fm \int \Fe(-\rho \xi) \, \widehat{f_1}(\xi) \,d\xi 
    \xrightarrow{\rho \to \infty} 0
  \end{split}
  \end{equation}
  by Lebesgue's theorem of dominated convergence:
  The Riemann-Lebesgue lemma \equ{RiemannLebesgue} tells that
  the constant function $0$ is the pointwise limit of
  the sequence of functions 
  $\Fe(-\rho \xi) \, \widehat{f_1}(\xi)$, $\rho \in \Z_+$, and
  \begin{equation}
    |\Fe(-\rho \xi) \, \widehat{f_1}(\xi) |
    \le \nl{\Fe}{\infty} \, |\widehat{f_1}(\xi) |
    \le \Fm \nl{e}{1} \, |\widehat{f_1}(\xi) | \in L^1.
  \end{equation}
  This completes the proof of claim (a).

  For claim (b), first choose $R > 0$ so large that $\supp e \subset B(0,R)$.
  Write $h = \chi_{B(0,1)}$ and 
  $h_\lambda (x) = \lambda^{-n} h(x / \lambda)$ for $\lambda > 0$.
  Then with $y = x/\rho$, $dy = \rho^{-n}\, dx$,
  \begin{equation}
  \begin{split}
    |\du{R_1 * f}{e_\rho}|
    &
%    = \left| \int_\Rn R_1 * f(x) \, e_\rho(x) \,dx \right|
%    \\&
    \le \int_\Rn |R_1 * f(x)| \, |e_\rho(x)| \,dx
    \\&
    \le \int_\Rn R_1 * |f| (x) \, |\rho^{-n} e(\pfrac{x}{\rho})| \,dx
    \\&
    =   \int_\Rn R_1 * |f| (\rho y) \, |e(y)| \,dy
    \\&
    \le \int_{B(0,R)} R_1 * |f| (\rho y) \, \nl{e}{\infty} \,dy
    \\&
    = \nl{e}{\infty} \int_\Rn R_1 * |f| (\rho y) \, h(\pfrac{y}{R}) \,dy
    \\&
    = \nl{e}{\infty} \int_\Rn R_1 * |f| (x) \,\rho^{-n} h(\pfrac{x}{R\rho})\,dx
    \\&
    = R^n \, \nl{e}{\infty} 
      \int_\Rn R_1 * |f| (x) \, (R\rho)^{-n} h(\pfrac{x}{R \rho}) \,dx
    \\&
    = R^n \, \nl{e}{\infty} \du{R_1*|f|}{h_{R\rho}},
  \end{split}
  \end{equation}
  so that it suffices to show that $\du{R_1*|f|}{h_{\tilde{\rho}}} \to 0$
  as $\tilde{\rho} = R\rho$ approaches infinity.

  To this end, first consider $f_1 \in \Dxr$ with 
  $f_1 \ge 0$ and $\supp f_1 \subset B(0,r_0)$.
  For $|x| \ge 2 r_0$ and $y \in \supp f_1$,
  \begin{equation}
    |x - y| \ge \big||x| - |y|\big| \ge |x| - \frac{|x|}{2} = \frac{|x|}{2}
  \end{equation}
  and $(1 + |y|)^{1-n} \ge (1 + r_0)^{1-n}$. Consequently,
  \begin{equation}
  \begin{split}
    \label{SKlemma314R1ConvF1EstimateForXGe2R0}
    R_1 * f_1(x)
    &
    = b_n \int_{\supp f_1} \frac{1}{|x - y|^{n-1}} \, |f_1(y)| \,dy
    \\&
    \le b_n \int_{\supp f_1} \left(\frac{2}{|x|}\right)^{n-1} \, 
        (1 + r_0)^{n-1} \, (1 + |y|)^{1-n} \, |f_1(y)| \,dy
    \\&
    = C_1 \, \nxr{f_1} \, |x|^{1-n} 
  \end{split}
  \end{equation}
  whenever $|x| > 2 r_0$, for some constant $C_1 > 0$ depending on 
  the size of the support of $f_1$.

  Now analyse 
  \begin{equation}
    \du{R_1 * f_1}{h_\rho}
    = \rho^{-n} \int_{|x| < \rho} R_1 * f_1(x) \, dx
  \end{equation}
  and split the range of integration into two parts,
  $|x| < 2 r_0$ and $2 r_0 \le |x| < \rho$, provided that $\rho > 2 r_0$.
  In the first one, use Lemma~\ref{thm:SKthm38} to see that
  \begin{equation}
  \begin{split}
    \rho^{-n} & \int_{|x| < 2 r_0} R_1 * f_1(x) \, dx
    \\&
    = \rho^{-n} \du{R_1 * f_1}{\chi_{B(0, 2 r_0)}}
    \\&
    \le \rho^{-n} C \, \nl{(1 + |x|)^{1-n} f_1}{1} 
      \left[\nl{\chi_{B(0, 2 r_0)}}{1} + 
            \nl{(1+|x|)^n \chi_{B(0, 2 r_0)}}{\infty}\right]
    \\&
    \le \rho^{1-n} C \, \nxr{f_1} \left[m(B(0,2 r_0)) + (1 + 2 r_0)^n\right],
  \end{split}
  \end{equation}
  if we assume that $\rho \ge 1$.
  In the second part, \equ{SKlemma314R1ConvF1EstimateForXGe2R0} shows than
  \begin{equation}
  \begin{split}
    \rho^{-n} \int_{2 r_0 \le |x| < \rho} R_1 * f_1(x) \, dx
    &
    \le \rho^{-n} \, C_1 \, \nxr{f_1} \, 
        \int_{2 r_0 \le |x| < \rho} |x|^{1-n} \,dx.
    \\&
    =   \rho^{-n} \, C_1 \, \nxr{f_1} \,|\Sm|
        \int_{2 r_0}^\rho r^{1-n} r^{n-1} \,dr
    \\&
    \le \rho^{1-n}\, C_1 \,\nxr{f_1} \,|\Sm|.
  \end{split}
  \end{equation}
  Therefore
  \begin{equation}
    \label{SKlemma314R1ConvF1HREstimate}
    |\du{R_1 * f_1}{h_\rho}|
    \le \frac{C_2 \, \nxr{f_1}}{\rho^{n-1}}
  \end{equation}
  when $\rho > 1$, for some constant $C_2 > 0$ depending on $f_1$.

  On the other hand, for any $f_2 \in \Dxr$ with $f_2|_{B(0,2)}=0$, 
  Lemma~\ref{lemma:Frostman} and the fact that 
  $1 + |y| \le 2 \, |y|$ whenever $|y| \ge 2$ show that
  \begin{equation}
  \begin{split}
    \label{SKlemma314R1ConvF2HREstimate}
    |\du{R_1 * f_2}{h_\rho}|
    &
    \le C_3 \, R_1 * f_2 (0)
    \\&
    =   C_3 \,b_n \int_{|y| \ge 2} |y|^{1-n} |f_2(y)| \,dy
    \\&
    \le C_3 \,b_n 
        \int_{|y| \ge 2} 2^{n-1}\, (1 + |y|)^{1-n} \,|f_2(y)| \,dy
    \\&
    \le C_4 \, \nxr{f_2}
  \end{split}
  \end{equation}
  for some constant $C_4 > 0$ independent of $f_2$.

  Now given any $\eps > 0$, Lemma~\ref{lemma:CInfty0IsDenseInDxr}
  shows that with $R \ge 2$ sufficiently large, $f_1 := \chi_{B(0,R)} \, |f|$
  satisfies $\nxr{|f| - f_1} < \eps/2 C_4$.
  This implies that with $f_2 = |f| - f_1$,
  \begin{equation}
    0 
    \le \du{R_1 * |f|}{h_\rho}
    = \du{R_1 * f_1}{h_\rho} + \du{R_1 * f_2}{h_\rho}
    \le \frac{C_2 \, \nxr{f_1}}{\rho^{n-1}}
      + C_4 \,\frac{\eps}{2 C_4}
    < \eps
  \end{equation}
  when $\rho$ is sufficiently large.
\end{tod}


The third auxiliary result is that if two distributions agree
on all test functions that vanish at the origin,
then their difference is a constant times Dirac's delta distribution.

\begin{lemma}
  \label{lemma:DistributionsAgreeForPhiVanishingAt0}
  Let $n \in \Z_+$. If $f, \, g \in \Ss'(\Rn)$ and
  \begin{equation}
    \du{f}{\phi} = \du{g}{\phi}
    \qquad
    \text{whenever }
    \phi(0) = 0,
  \end{equation}
  then $f = g + c \delta$ for some constant $c \in \C$.
\end{lemma}

\begin{tod}
  First show that $\supp (f - g) \subset \{0\}$.
  Let $y \ne 0$ and choose an open set $U \subset \Rn$ with
  $y \in U$, $0 \not\in U$.
  Then $\phi(0) = 0$ for all $\phi \in \DD$ with $\supp \phi \subset U$,
  so that by assumption $\du{f - g}{\phi} = 0$.
  In other words, every point $y \ne 0$ has a neighbourhood
  on which $f - g$ vanishes, and therefore $y \not\in \supp (f - g)$.

  Since $f - g$ has point support, it must be a finite linear combination
  of Dirac's delta distribution and its derivatives:
  \begin{equation}
    f - g = \sum_{|\alpha| \le m} c_\alpha \, D^\alpha \delta.
  \end{equation}
  It suffices to show that 
  $c_0$ must be the only non-zero one of the constants $c_\alpha$.
  Assume contrariwise that $c_\beta \ne 0$ and $\beta \ne (0,0,\ldots,0)$.
  Choose a test function $\psi$ such that $\partial^\beta \psi(0) \ne 0$
  and $D^\alpha \psi(0) = 0$ for all $\alpha \ne \beta$;
  especially $\psi(0) = 0$. 
  Such a function could be, for example, $x^\beta \chim_{1,2}$.
  This leads to the contradiction
  \begin{equation}
    0 
    = \du{f - g}{\psi} 
    = \sum_{|\alpha| \le m} c_\alpha \du{D^\alpha \delta}{\psi}
    = \sum_{|\alpha| \le m} c_\alpha \partial^\alpha \psi(0)
    = c_\beta \partial^\beta \psi(0)
    \ne 0.
  \end{equation}
\end{tod}



We are now ready to extend to $\Dxr$ the property that $R_1 * g$
corresponds on the Fourier transformed side to $|\xi|^{-1} \, \Fg$.

\begin{theorem}
  \label{thm:FourierTransformOfRiestPotentialInDxr}
  Assume that $n \ge 2$,
  $f \in \Dxr(\Rn)$ and $\phi \in \Ss(\Rn)$ with $\phi(0) = 0$.
  Then
  \begin{equation}
    \du{\F(R_1 * f)}{\phi} = \int |\xi|^{-1}\, \Ff(\xi) \, \phi(\xi) \,d\xi.
  \end{equation}
  If, in addition, $|\xi|^{-1} \Ff(\xi) \in \Lloc(\Rn)$, then
  \begin{equation}
    \F(R_1 * f)(\xi) = |\xi|^{-1} \Ff(\xi).
  \end{equation}
\end{theorem}

\begin{tod}
  First construct functions $\phi_j \in \Ss'$ such that
  $\phi(x) = \sum_{j=1}^n x_j \, \phi_j(x)$, as follows.

  Define the functions \cite{faridani92:local}
  \begin{equation}
    \setlength{\extrarowheight}{0.5ex} \begin{array}{r@{\,}l@{\,}r@{\,}l}
      \phi^- &= \chim_{2,3} \phi \in \Ss 
      & \qquad
         a^- &= \chim_{3,4} \in C^\infty_0 
      \\
      \phi^+ &= \phi - \phi^- \in \Ss
      &\qquad
         a^+ &= 1 - \chim_{1,2} \in C^\infty
     \end{array} 
   \end{equation}
  where $\chim_{\eps_1,\eps_2}$ is the approximate 
  characteristic function of \equ{ApproxCharacteristicFunction}.
  They have the properties
  \begin{equation} 
    \begin{array}{r@{\,}l@{\,}r@{\,}l}
      \phi^-(x) &= 
      \begin{cases}
        \phi(x), & |x| \le 2 \\ 0, & |x| \ge 3 
      \end{cases}
      & \qquad
         a^-(x) &=
      \begin{cases}
        1, & |x| \le 3 \\ 0, & |x| \ge 4 
      \end{cases}
      \\ \\
      \phi^+(x) &=
      \begin{cases}
        0, & |x| \le 2 \\ \phi(x), & |x| \ge 3
      \end{cases}
      &\qquad
         a^+(x) &=
      \begin{cases}
        0, & |x| \le 1 \\ 1, & |x| \ge 2. 
      \end{cases}
    \end{array}
  \end{equation}
  Integrating along the straight line from $0$ to $x$ gives
  \begin{equation}
    \phi^-(x) = \int_0^1 \nabla \phi^-(tx) \cdot x \,dt
    = \sum_{j = 1}^n x_j \int_0^1 \dodo{\phi^-}{x_j} (tx) \,dt,
  \end{equation}
  and along the ray starting from $x$ and going away from the origin,
  \begin{equation}
    \phi^+(x) = -\int_1^\infty \nabla \phi^+(tx) \cdot x \,dt
    = - \sum_{j = 1}^n x_j \int_1^\infty \dodo{\phi^+}{x_j} (tx) \,dt.
  \end{equation}
  Note that
  \begin{equation}
    \phi^\pm(x) = a^\pm(x) \, \phi^\pm(x) = \sum_{j=1}^n x_j \, \phi_j^\pm(x),
  \end{equation}
  where
  \begin{equation}
    \phi_j^-(x) = a^-(x) \int_0^1 \dodo{\phi^-}{x_j} (tx) \,dt
    \quad \text{and} \quad
    \phi_j^+(x) = - a^+(x) \int_1^\infty \dodo{\phi^+}{x_j} (tx) \,dt
  \end{equation}
  are rapidly decreasing functions: 
  For $\phi_j^-$, this follows from the estimate
  \begin{equation}
  \begin{split}
    \norm[\alpha,\beta]{\phi_j^-}
    &
    = \sup_{x \in \Rn} \Big| \, x^\alpha \sum_{\gamma \le \beta} 
      c_{\beta \gamma} \partial^\gamma a^-(x) 
      \int_0^1 \partial^{\beta - \gamma + e_j} \phi^-(tx) \,dt \, \Big|
    \\&
    \le \sup_{x \in \Rn} \sum_{\gamma \le \beta} c_{\beta \gamma}
      \left| x^\alpha \partial^\gamma a^-(x) \right|
      \int_0^1 \left| \partial^{\beta - \gamma + e_j} \phi^-(tx) \right| \,dt
    \\&
    \le \sum_{\gamma \le \beta} c_{\beta \gamma} \norm[\alpha,\gamma]{a^-}
      \int_0^1 \norm[0,\beta - \gamma + e_j]{\phi^-} \,dt
    < \infty.
  \end{split}
  \end{equation}
  For $\phi_j^+$,
  \begin{equation}
  \begin{split}
    \norm[\alpha,\beta]{\phi_j^+}
    &
    = \sup_{x \in \Rn} \Big| \, x^\alpha \sum_{\gamma \le \beta}
      c_{\beta \gamma} \partial^\gamma a^+(x) 
      \int_1^\infty \partial^{\beta - \gamma + e_j} \phi^+(tx) \,dt \, \Big|
    \\&
    \le \sup_{x \in \Rn} \sum_{\gamma \le \beta} c_{\beta \gamma}
      \left| \partial^\gamma a^+(x) \right|
      \int_1^\infty |x^\alpha| \, |tx|^2 \, 
        \left|\partial^{\beta - \gamma + e_j} \phi^+(tx) \right| 
        \, |x|^{-2} \, t^{-2} \,dt.
  \end{split}
  \end{equation}
  Note that for $\phi_j^+(x)$ to be non-zero, $|x|$ must be greater than 2,
  and that since $t \ge 1$ in the integral, $|x^\alpha| \le |(tx)^\alpha|$.
  As $|tx|^2 = \sum_{k = 1}^n (tx)^{2e_k}$, this yields
  \begin{equation}
  \begin{split}
    \norm[\alpha,\beta]{\phi_j^+}
    &
    \le \sup_{x \in \Rn} \sum_{\gamma \le \beta} c_{\beta \gamma}
      \left| \partial^\gamma a^+(x) \right|
      \int_1^\infty \left|(tx)^{\alpha + 2e_k}
        \partial^{\beta - \gamma + e_j} \phi^+(tx) \right| \, t^{-2} \,dt
    \\&
    \le \sum_{\gamma \le \beta} \sum_{k=1}^n c_{\beta \gamma}
      \norm[0,\gamma]{a^+} \, \norm[\alpha + 2e_k, \beta-\gamma+e_j]{\phi^+}
      \int_1^\infty t^{-2} \,dt
    < \infty,
  \end{split}
  \end{equation}
  because although $a^+ \not\in \Ss$, all of its derivatives are
  constant outside $B(0,2)$, so that $\norm[0, \gamma]{a^+} < \infty$
  for all $\gamma \in \N^n$.

  This gives us the desired functions $\phi_j$:
  \begin{equation}
    \phi(x) = \phi^-(x) + \phi^+(x) = \sum_{j=1}^n x_j \, \phi_j(x),
    \qquad
    \text{where}
    \quad
    \phi_j = \phi_j^- + \phi_j^+ \in \Ss.
  \end{equation}

  Now $R_1 * f \in \Lloc \subset \Ss'$ by Theorem~\ref{thm:SKthm38}, 
  so that Theorem~\ref{thm:DerivativeOfRieszPotential} shows that
  \begin{equation}
    \xi_j \F(R_1 * f) 
    = -i \, \F\left(D_j (R_1 * f) \right)
    = -i \, \F\left(\pv \dodo{R_1}{x_j} * f \right)
    = \frac{\xi_j}{|\xi|} \, \Ff(\xi).
  \end{equation}
  Therefore
  \begin{equation}
  \begin{split}
    \du{\F(R_1 * f)}{\phi}
    &
    = \sum_{j=1}^n \du{\F(R_1 * f)}{\xi_j \phi_j}
    \\&
    = \sum_{j=1}^n \du{\xi_j \, \F(R_1 * f)}{\phi_j}
    \\&
    = \sum_{j=1}^n \Du{\frac{\xi_j}{|\xi|}\, \Ff}{\phi_j}
    \\&
    = \int |\xi|^{-1} \, \Ff(\xi) \sum_{j=1}^n \xi_j \, \phi_j(\xi) \,d\xi
    \\&
    = \int |\xi|^{-1} \, \Ff(\xi) \, \phi(\xi) \, d\xi,
  \end{split}
  \end{equation}
  which proves the first claim.

  Then assume that $|\xi|^{-1} \Ff \in \Lloc \subset \Ss'$, so that
  \begin{equation}
    \du{\F(R_1 * f)}{\psi}
    = \du{|\xi|^{-1} \Ff}{\psi}
  \end{equation}
  for all $\psi \in \Ss$ with $\psi(0) = 0$.
  Lemma~\ref{lemma:DistributionsAgreeForPhiVanishingAt0} then shows that
  $\F(R_1 * f) = |\xi|^{-1} \Ff + c \delta$ for some constant $c \in \C$.
  It remains to show that $c = 0$.

  Write $u = \Fi(|\xi|^{-1} \Ff)$.
  Since $\Fu$ was assumed to be locally integrable and
  \begin{equation}
    \nl{\chi_{\Rn \setminus B(0,1)} |\xi|^{-1} \Ff}{2}
    \le \nl{\Ff}{2} = \nl{f}{2} \le \nxr{f} < \infty,
  \end{equation}
  we see that
  \begin{equation}
    \Fu = \chi_{B(0,1)} \Fu + \chi_{\Rn \setminus B(0,1)} \Fu \in L^1 + L^2.
  \end{equation}
  Thus for any $e \in C^\infty_0$ with $\int e(x) \,dx = 1$, and
  $e_\rho(x) = \rho^{-n} e(x/\rho)$,
  \begin{equation}
    c = \du{c}{e_\rho} 
    = \du{\Fi(c \delta)}{e_\rho}
    = \du{R_1 * f}{e_\rho} - \du{u}{e_\rho}
    \xrightarrow{\rho \to \infty} 0
  \end{equation}
  by Lemma~\ref{lemma:SKlemma314}. This completes the proof.
\end{tod}

Note that the condition that $|\xi|^{-1} f(\xi)$ be locally integrable
in Theorem~\ref{thm:FourierTransformOfRiestPotentialInDxr}
is always true if $n > 2$, because then
\begin{equation}
\begin{split}
  \nl{\chi_{B(0,R)}(\xi) \, |\xi|^{-1} \Ff(\xi)}{1}
  &
  \le \nl{\chi_{B(0,R)}(\xi) \, |\xi|^{-1}}{2} \nl{\Ff}{2}
  \\&
  \le \sqrt{|\Sm| \int_0^R \frac{r^{n-1} \,dr}{r^2}} \, \nl{f}{2}
  \\&
  \le \sqrt{|\Sm| \sij{0}{R} r^{n-2}} \, \nxr{f}
  < \infty,
\end{split}
\end{equation}
or if $f \in L^p$, $p > 2$, 
because then $p' < 2$ and $n - 1 - p' \ge 1 - p' > -1$, and consequently
\begin{equation}
\begin{split}
  \nl{\chi_{B(0,R)}(\xi) \, |\xi|^{-1} \Ff(\xi)}{1}
  &
  \le \nl{\chi_{B(0,R)}(\xi) \, |\xi|^{-1}}{p'} \nl{\Ff}{p}
  \\&
  = |\Sm| \left(\int_0^R r^{n-1-p'} \,dr \right)^{\frac{1}{p'}} \nl{f}{p} 
  < \infty.
\end{split}
\end{equation}



\section{Properties of the Radiograph Operators}
\label{sec:PropertiesOfRadiographOperators}

In this section, we shall investigate some properties of the
divergent beam radiograph operator $\DD$ and the
parallel beam radiograph operator $\PP$, introduced in
Definitions \ref{def:DivBeamRadiograph} and \ref{def:ParBeamRadiograph}.

The Riesz potential of an x-ray attenuation density can be calculated
from the divergent or parallel beam radiographs:

\begin{lemma}
  \label{lemma:SKthm27}
  If $n \ge 2$ and $f \in \Dxr(\Rn)$, then for almost all $x \in \Rn$,
  \begin{equation}
    \label{SKthm27claim}
    \frac{1}{b_n} \, R_1 * f(x) 
    = \int_\Sm \DD_x f(\theta) \,d\theta
    = \frac{1}{2} \int_\Sm \PP_\theta f(E_\theta x) \,d\theta
  \end{equation}
  and the integrals converge absolutely.

  Equation \equ{SKthm27claim} also holds
  for all non-negative, measurable functions $f$,
  in the sense that when one of the members is infinite, so are the other two,
  and otherwise all three take the same finite value.
\end{lemma}

\begin{tod}
  If $f \in \Dxr$, the changes of variable $t = |y|$, $\theta = y/|y|$, 
  $dy = t^{n-1} \,dt \,d\theta$ and $y' = -y$ yield
  \begin{equation}
  \begin{split}
    \label{SKthm27equality1}
    \int_\Sm \DD_x f(\theta) \, d\theta
    &
    = \int_\Sm \int_0^\infty f(x + t\theta) \,dt \,d\theta
    \\&
    = \int_\Rn f(x + y) \, |y|^{1-n} \,dy
    \\&
    = \int_\Rn f(x - y') \, |y'|^{1-n} \,dy'
    \\&
    = \frac{1}{b_n} \, R_1 * g(x),
  \end{split}
  \end{equation}
  for almost all $x \in \Rn$ by Fubini's theorem, 
  since Theorem~\ref{thm:SKthm38} asserts 
  the absolute convergence of the integral.
  For the second equality in \equ{SKthm27claim}, observe that 
  $\PP_\theta f(E_\theta x) = \DD_x f(\theta) + \DD_x f(-\theta)$,
  whence with $\theta' = -\theta$,
  \begin{equation}
  \begin{split}
    \label{SKthm27equality2}
    \frac{1}{2} \int_\Sm \PP_\theta f(E_\theta x) \, d\theta
    &
    = \frac{1}{2} \left[ 
      \int_\Sm \DD_x f(\theta) \,d\theta + \int_\Sm \DD_x f(-\theta) \,d\theta
      \right]
    \\&
    = \frac{1}{2} \left[ 
      \int_\Sm \DD_x f(\theta) \,d\theta + \int_\Sm \DD_x f(\theta') \,d\theta'
      \right]
    \\&
%    = \int_\Sm \DD_x f(\theta) \, d\theta
    = \frac{1}{b_n} \, R_1 * f(x).
  \end{split}
  \end{equation}
  If then $f \ge 0$, \equ{SKthm27equality1} is justified by
  Fubini's theorem for non-negative functions, and
  \equ{SKthm27equality2} holds equally well.
\end{tod}


That $\Dxr$ is a reasonable domain for x-ray attenuation densities,
as asserted by Theorem~\ref{thm:SKthm38}, 
can be formulated in terms of the divergent beam radiograph as follows:

\begin{corollary}
  \label{cor:SKthm41}
  If $n \ge 2$ and $f \in \Dxr(\Rn)$, 
  then for almost all $x \in \Rn$, $\DD_x f$
  is defined almost everywhere on $\Sm$ by an absolutely convergent integral
  and $\DD_x f \in L^1(\Sm)$.

  If, on the contrary, $(1 + |x|)^{1-n} f(x) \not\in L^1(\Rn)$ and $f \ge 0$,
  then $\DD_x f \not\in L^1(\Sm)$ for any $x \in \Rn$.
\end{corollary}

\begin{tod}
  By the triangle inequality and Lemma~\ref{lemma:SKthm27},
  \begin{equation}
%  \begin{split}
    \int_\Sm \left|\DD_x f(\theta)\right| \,d\theta
%    &
%    = \int_\Sm \left| \int_0^\infty f(x + t \theta) \,dt \right| \,d\theta
%    \\&
%    \le \int_\Sm \int_0^\infty |f(x + t \theta)| \,dt \,d\theta
    \le \int_\Sm \DD_x |f|(\theta) \,d\theta
    = \frac{1}{b_n} \, R_1 * |f|(x),
%  \end{split}
  \end{equation}
  which converges for almost all $x \in \Rn$ by Theorem~\ref{thm:SKthm38}.
  For such $x$, the absolute value integral
  \begin{equation}
    \DD_x |f|(\theta) = \int_0^\infty |f(x + t \theta)| \,dt
  \end{equation}
  must be finite for almost every $\theta \in \Sm$.

  If then $f \ge 0$ and $(1 + |x|)^{1-n} f(x) \not\in L^1(\Rn)$,
  \begin{equation}
    \int_\Sm \DD_x f(\theta) \,d\theta = \frac{1}{b_n} R_1 * f(x) = \infty
  \end{equation}
  for all $x \in \Rn$, again by Theorem~\ref{thm:SKthm38}.
\end{tod}


For the parallel beam radiograph, we have the following result:
%\kommentti{\cite[Theorem 2.7]{smith85:mathfound}:sta muuten 
%  puuttuu tästä vakio $2/b_n$.}

\begin{lemma}
  \label{lemma:SKthm42}
  Let $n \ge 2$ and $f \in \Dxr(\Rn)$. Then for almost all $\theta \in \Sm$,
  $\PP_\theta f$ is defined almost everywhere on $\theta^\bot$
  by an absolutely convergent integral.

  If $\phi \in L^1(\Rn)$ and $(1 + |x|)^n \phi \in L^\infty(\Rn)$, then
  \begin{equation}
    \label{SKthm42claim2}
    \int_\Sm \du{\PP_\theta f}{\PP_\theta \phi} \,d\theta 
    = \frac{2}{b_n} \, \du{R_1 * f}{\phi}
  \end{equation}
  and
  \begin{equation}
    \label{SKthm42claim3}
    \Big| \int_\Sm \du{\PP_\theta f}{\PP_\theta \phi} \,d\theta \, \Big|
    \le C \, \nxr{f} 
      \Big[ \nl{\phi}{1} + \Nl{(1 + |x|)^n \phi}{\infty} \Big]
  \end{equation}
  for some constant $C$.
\end{lemma}

\begin{tod}
  By Lemma~\ref{lemma:SKthm27},
  \begin{equation}
  \begin{split}
    \label{SKthm42FirstIntegrals}
    \frac{2}{b_n} \, \du{R_1 * f}{\phi}
    &
    = \Du{\int_\Sm \PP_\theta f(E_\theta x) \,d\theta}{\phi}
    \\&
    = \int_\Rn \int_\Sm \infint f(x + t\theta) \,dt \,d\theta \, \phi(x) \,dx
    \\&
    = \int_\Sm \int_\Rn \infint f(x + t\theta) \,dt \, \phi(x) \,dx \,d\theta.
  \end{split}
  \end{equation}
  The changes of variable 
  $x = x' + s \theta \in \theta^\bot \oplus \R \theta = \Rn$, 
  $dx = dx' \, ds$ and $t' = s + t$, $dt' = dt$ yield
  \begin{equation}
  \begin{split}
    \frac{2}{b_n} \, \du{R_1 * f}{\phi}
    &
    = \int_\Sm \int_{\theta^\bot} \infint \infint 
      f(x' + s\theta + t\theta) \,dt \, \phi(x' + s\theta) \,ds \,dx' \,d\theta
    \\&
    = \int_\Sm \int_{\theta^\bot} \infint f(x' + t'\theta) \,dt' 
        \infint \phi(x' + s\theta) \,ds \,dx' \,d\theta
    \\&
    = \int_\Sm \du{\PP_\theta f}{\PP_\theta \phi} \,d\theta.
  \end{split}
  \end{equation}
  The order of integration can be changed above by Fubini's theorem,
  since the integral in \equ{SKthm42FirstIntegrals} converges 
  absolutely: By Lemma~\ref{lemma:SKthm27} and Theorem~\ref{thm:SKthm38},
  \begin{equation}
  \begin{split}
    \int_\Rn \int_\Sm \infint |f(x & + t\theta)| \,dt\,d\theta\, |\phi(x)| \,dx
    \\&
    = \frac{2}{b_n} \, \du{R_1 * |f|}{|\phi|}
    \\&
    \le C \nl{(1 + |x|)^{1-n} f}{1} 
      \left[ \nl{\phi}{1} + \nl{(1 + |x|)^n \phi}{\infty} \right]
    \\&
    \le C \nxr{f} \Big[ \nl{\phi}{1} + \nl{(1 + |x|)^n \phi}{\infty} \Big]
    < \infty
  \end{split}
  \end{equation}
  for some constant $C > 0$.
  This also proves \equ{SKthm42claim3} because 
  \begin{equation}
  \begin{split}
    \Big| \int_\Sm \du{\PP_\theta f}{\PP_\theta \phi} \,d\theta \, \Big|
    &
    = \Big|\int_\Rn \int_\Sm \infint f(x + t\theta) \,dt \,d\theta \, 
        \phi(x) \,dx \, \Big|
    \\&
    \le \int_\Rn \int_\Sm \infint 
          |f(x + t\theta)| \,dt \,d\theta \, |\phi(x)| \,dx.
  \end{split}
  \end{equation}
  For showing the absolute convergence of $\PP_\theta f(x)$ 
  almost everywhere, fix $\phi(x) = e^{-|x|^2}$.
  As $|f|$ satisfies the conditions of the theorem as well as $f$,
  \begin{equation}
    \int_\Sm \du{\PP_\theta |f|}{\PP_\theta \phi} < \infty
  \end{equation}
  by \equ{SKthm42claim3}.
  The integrand $\du{\PP_\theta |f|}{\PP_\theta \phi}$ 
  must therefore be finite for almost all $\theta \in \Sm$.

  For such $\theta$, the integral $\PP_\theta f(x)$ must converge absolutely
  almost everywhere, for if this were not the case,
  then there would be a set $E \subset \theta^\bot$ with
  $m(E) > 0$ and $\PP_\theta |f|(x) = \infty$ for all $x \in E$.
  By the regularity of the Lebesgue measure, $E$ could be chosen compact,
  so that $M := \inf_{x \in E} \PP_\theta \phi(x) > 0$, and consequently
  \begin{equation}
    \du{\PP_\theta |f|}{\PP_\theta \phi}
    \ge \int_E \PP_\theta |f|(x) \, \PP_\theta \phi(x) \,dx
    \ge M \int_E \PP_\theta |f|(x) \,dx = \infty.
  \end{equation}
\end{tod}


The following, so called projection slice theorem
shows how the Fourier transform of the parallel beam radiograph
can be obtained from the Fourier transform of the
x-ray attenuation density:

\begin{lemma}
  \label{lemma:ProjectionSliceTheorem}
  If $f \in L^1(\Rn)$ and $\theta \in \Sm$, then
  $\PP_\theta f \in L^1(\theta^\bot)$ and
  \begin{equation}
    \F \left(\PP_\theta f\right) (\xi) = \sqrt{2 \pi}\, \Ff(\xi)
  \end{equation}
  for all $\xi \in \theta^\bot$.
\end{lemma}

\begin{tod}
  Obviously,
  \begin{equation}
    \int_{\theta^\bot} |\PP_\theta f(x)| \,dx
    \le \int_{\theta^\bot} \infint |f(x + t\theta)| \,dt\,dx
    = \nl{f}{1} 
    < \infty.
  \end{equation}
  For $f \in L^1$, the change of variable 
  $y = x + t\theta \in \theta^\bot \oplus \R \theta = \Rn$,
  $dy = dx \, dt$ yields, after noting that 
  $\xi \in \theta^\bot$ implies 
  $x \cdot \xi = x \cdot \xi + t \theta \cdot \xi = y \cdot \xi$,
  \begin{equation}
  \begin{split}
    \F\left(\PP_\theta f\right)(\xi)
    &
    = (2 \pi)^{-\frac{n-1}{2}} \int_{\theta^\bot} \infint f(x + t\theta)
      e^{-i x \cdot \xi} \,dt \,d\xi
    \\&
    = \sqrt{2 \pi} \, \Fm \int_\Rn f(y) \, e^{-i y \cdot \xi} \,dx
    = \sqrt{2 \pi} \, \Ff(\xi).
  \end{split}
  \end{equation}
% If then $f \in L^2$, choose a sequence of functions $f_k \in L^1 \cap L^2$
%  with $\nl{f - f_k}{2} \to 0$. Then
%  \begin{equation}
%  \begin{split}
%    \norm[L^2(\thb)]{\F\PP_\theta f - \sqrt{2\pi} \, \Ff}
%    &
%    \le \norm[L^2(\thb)]{\F\PP_\theta f - \F\PP_\theta f_k}
%      + \norm[L^2(\thb)]{\sqrt{2\pi} \, \Ff_k - \Ff}
%    \\&
%    = \norm[L^2(\thb)]{\PP_\theta (f - f_k)}
%      + \sqrt{2\pi} \, \norm[L^2(\thb)]{(f - f_k)}
%    \xrightarrow{k \to \infty} 0
%  \end{split}
%  \end{equation}
\end{tod}


The following result will also be needed:

\begin{lemma}
  \label{lemma:SKlemma46}
  If $h: \Rn \to [0, \infty)$ is measurable, then
  \begin{equation}
    \int_\Sm \int_{\zeta^\bot} |y| \, h(y) \, dy \, d\zeta
    = |S^{n-2}| \int_\Rn h(x) \,dx.
  \end{equation}
\end{lemma}

\begin{tod}
  First observe that $\Sm \cap \theta^\bot \cong S^{n-2}$, whence
%  $m_{n-2}(\Sm \cap \theta^\bot) = |S^{n-2}|$ 
%  for all $\theta \in \Sm$, which can be seen as follows: 
%  Choose coordinates in such a way that $\theta = e_1$
%  and let $x \in \Sm \cap \theta^\bot$.
%  Then
%  \begin{equation}
%    0 = x \cdot \theta = \sum_{k=1}^n x_k \, e_k = x_1
%    \quad \text{and} \quad
%    1 = \sum_{k=1}^n x_k^2 = \sum_{k=2}^n x_k^2.
%  \end{equation}
%  Also conversely, $x_1 = 0$, $\sum_{k=2}^n x_k^2 = 1$ imply
%  that $x \in \Sm \cap \theta^\bot$.
%  Therefore $\Sm \cap \theta^\bot$ is the unit sphere of 
%  $\{0\} \times \R^{n-1}$.
%
%  This shows that 
  for $\hm: \Sm \to [0, \infty)$,
  \begin{equation}
%  \begin{split}
    \int_\Sm \int_{\Sm \cap \theta^\bot} \hm(\theta)\,  d\zeta \, d\theta
%    &
%    = \int_\Sm \int_{\Sm \cap \theta^\bot} d\zeta \,\hm(\theta) \, d\theta
%    \\&
    = |S^{n-2}| \int_\Sm \hm(\theta) \,d\theta.
%  \end{split}
  \end{equation}
  Consequently for $h: \Rn \to [0, \infty)$,
  \begin{equation}
  \begin{split}
    |S^{n-2}| \int_\Rn h(x) \,dx
    &
    = |S^{n-2}| \int_0^\infty \int_\Sm h(r \theta) \,d\theta \, r^{n-1} \,dr
    \\&
    = \int_0^\infty \int_\Sm \int_{\Sm \cap \theta^\bot} h(r\theta) 
        \,d\zeta \,d\theta \, r^{n-1} \,dr.
  \end{split}
  \end{equation}
  As
  \begin{equation}
  \begin{split}
    \theta \in \Sm \; \wedge \; \zeta \in \Sm \cap \theta^\bot 
    &
    \quad \ekv \quad
    \theta \in \Sm \;\wedge\; \zeta \in \Sm \;\wedge\; \theta \cdot \zeta = 0 
    \\&
    \quad \ekv \quad
    \zeta \in \Sm \; \wedge \; \theta \in \Sm \cap \zeta^\bot,
  \end{split}
  \end{equation}
  the change of variable $y = r \theta \in \zeta^\bot$,
  $dy = r^{n-2} \,dr \,d\theta$ gives
  \begin{equation}
  \begin{split}
    |S^{n-2}| \int_\Rn h(x) \,dx
    &
    = \int_0^\infty \int_\Sm \int_{\Sm \cap \zeta^\bot} h(r\theta) 
        \,d\theta \,d\zeta \, r^{n-1} \,dr
    \\&
    = \int_\Sm \int_{\Sm \cap \zeta^\bot} \int_0^\infty
      r \, h(r \theta) \, r^{n-2} \, dr \, d\theta \, d\zeta
    \\&
    = \int_\Sm \int_{\zeta^\bot} |y| \, h(y) \, dy \, d\zeta.
  \end{split}
  \end{equation}
\end{tod}


The projection slice theorem (Lemma~\ref{lemma:ProjectionSliceTheorem})
can now be extended to cover certain functions in the x-ray domain:

\begin{theorem}
  \label{thm:SKthm48}
  \samepage
  Let $f \in \Dxr(\Rn)$. Then for almost all $\theta \in \Sm$,
  \vspace{-2ex}
  \begin{enumerate}
  \item[(a)] $|\xi|^{1/2} \Ff \in L^2(\theta^\bot)$
  \item[(b)] $\PP_\theta f \in \Lloc(\theta^\bot) \subset \Ss'(\theta^\bot)$
  \item[(c)] if $\phi \in \Ss(\theta^\bot)$ and $\phi(0) = 0$, then
    \begin{equation}
      \Du{\F\left(\PP_\theta f\right)}{\phi} 
      = \sqrt{2\pi} \int_{\theta^\bot} \Ff(\xi) \, \phi(\xi) \,d\xi
    \end{equation}
  \item[(d)] if $|\xi|^{-1} \Ff(\xi) \in \Lloc(\Rn)$, then
    $\Ff \in \Lloc(\theta^\bot)$ and
    \begin{equation}
      \F\left(\PP_\theta f\right)(\xi) = \sqrt{2\pi} \, \Ff(\xi)
    \end{equation}
    for almost all $\xi \in \theta^\bot$.
  \end{enumerate}
\end{theorem}

\begin{tod}
  Claim (a) follows from Lemma~\ref{lemma:SKlemma46}:
  \begin{equation}
    \label{SKequ49}
    \int_\Sm \int_{\theta^\bot} |\xi| \, |\Ff(\xi)|^2 \,d\xi \, d\theta
    = |\Sm| \int_\Rn |\Ff(\xi)|^2 \,d\xi
    = |\Sm| \nl{f}{2}^2
    < \infty,
  \end{equation}
  whence
  \begin{equation}
    \norm[L^2(\theta^\bot)]{|\xi|^{1/2} \Ff}^2
    = \int_{\theta^\bot} |\xi| \, |\Ff(\xi)|^2 \,d\xi
    < \infty
  \end{equation}
  for almost all $x \in \theta^\bot$.

  For claim (b), we show that for almost all $\theta \in \Sm$,
  \begin{equation}
    \int_{B(0,R) \cap \theta^\bot} |\PP_\theta f(x)| \,dx
    < \infty
  \end{equation}
  for any $R > 0$.
  Note that
  \begin{equation}
  \begin{split}
    \int_{B(0,R) \cap \theta^\bot} |\PP_\theta f(x)| \,dx
    &
    \le \int_{B(0,R) \cap \theta^\bot} |\PP_\theta f(x)| \,
          \frac{(1 + R)^n}{(1 + |x|)^n} \,dx
    \\&
    \le (1 + R)^n \int_{\theta^\bot} 
        |\PP_\theta f(x)| \, (1 + |x|)^{-n} \,dx
  \end{split}
  \end{equation}
  and use the estimate
  \begin{equation}
  \begin{split}
    \PP_\theta\left((1 + |x|)^{-n-1}\right)(x)
    &
    = \infint (1 + |x + t \theta|)^{-n-1} \,dt
    \\&
    \ge \infint (1 + |x| + |t|)^{-n-1} \,dt
    \\&
    = 2 \int_0^\infty (1 + |x| + t)^{-n-1} \,dt
    \\&
    = \frac{2}{n} \, (1 + |x|)^{-n}
  \end{split}
  \end{equation}
  to see that 
  \begin{equation}
    \int_{B(0,R) \cap \theta^\bot} |\PP_\theta f(x)| (1 + |x|)^{-n} \,dx
    \le \frac{n}{2} \, (1 + R)^n \,
      \Du{|\PP_\theta f(x)|}{\PP_\theta\left((1 + |x|)^{-n-1}\right)}.
  \end{equation}
  Lemma~\ref{lemma:SKthm42} shows that for some constant $C_1 > 0$,
  \begin{multline}
    \label{SKequ410}
%  \begin{equation}
%  \begin{split}
    \int_\Sm \int_{B(0,R) \cap \theta^\bot} 
      |\PP_\theta f(x)| \, (1 + |x|)^{-n} \,dx \,d\theta
%    \\&
%    \le \frac{n}{2} \int_\Sm 
%      \Du{|\PP_\theta f(x)|}{\PP_\theta\left((1 + |x|)^{-n-1}\right)} 
%      \,d\theta
%    \\&
    \\
    \le C_1 \, \nxr{f} \, \Big[
      \Nl{(1 + |x|)^{-n-1}}{1} + \Nl{(1 + |x|)^n (1 + |x|)^{-n-1}}{\infty}
      \Big],
%  \end{split}
%  \end{equation}
  \end{multline}
  which is finite. This proves (b).

  For claim (c), choose a sequence of functions $f_k \in L^1 \cap \Dxr$
  converging to $f$ in the $\Dxr$ norm; 
  this is possible since the subset $C^\infty_0$ of $L^1 \cap \Dxr$ is
  dense by Lemma~\ref{lemma:CInfty0IsDenseInDxr}.
  Then
  \begin{equation}
    \int_\Sm \int_{\theta^\bot} 
      |\PP_\theta f(x) - \PP_\theta f_k(x)| \, (1 + |x|)^{-n} \,dx \,d\theta
    \le C_2 \, \nxr{f - f_k}
    \xrightarrow{k \to \infty} 0
  \end{equation}
  by \equ{SKequ410}.
  This convergence in the $L^1(\Sm)$ norm
  implies that there is a subsequence $(f_{k_l})_{l \in \N}$
  such that
  \begin{equation}
    \int_{\theta^\bot} 
      |\PP_\theta f(x) - \PP_\theta f_{k_l}(x)| \, (1 + |x|)^{-n} \,dx
    \xrightarrow{l \to \infty} 0
  \end{equation}
  for almost all $\theta \in \Sm$. 
  For such $\theta$ and any $\psi \in \Ss(\theta^\bot)$,
  \begin{equation}
  \begin{split}
    \label{SKthm48cPthetafklToPthetafInSPrime}
    \Du{\PP_\theta f - \PP_\theta f_{k_l}}{\psi}
    &
    \le \int_{\theta^\bot} \left|\PP_\theta f(x) - \PP_\theta f_{k_l} \right|
      \, (1 + |x|)^{-n} \, \left|(1 + |x|)^n \psi(x) \right| \,dx
    \\&
    \le C_3 \int_{\theta^\bot} 
      \left|\PP_\theta f(x) - \PP_\theta f_{k_l} \right| \, (1 + |x|)^{-n} \,dx
    \xrightarrow{l \to \infty} 0,
  \end{split}
  \end{equation}
  for some $C_3 > 0$ depending on the seminorms
  $\norm[\alpha, 0]{\psi}$, $|\alpha| \le 2n$.
%  Thus $\PP_\theta f_{k_l} \to \PP_\theta f$ in $\Ss'(\theta^\bot)$
%  for almost all $\theta \in \Sm$.
%
  Lemma~\ref{lemma:ProjectionSliceTheorem} therefore tells us that
  \begin{equation}
  \begin{split}
    \label{SKthm38cIntFfklPsiToDuFPThetaFPsi}
    \sqrt{2\pi} \int_{\theta^\bot} \widehat{f_{k_l}}(\xi) \,\phi(\xi) \,d\xi
    &
%    = \int_{\theta^\not} 
%      \F\left(\PP_\theta f_{k_l}\right)(\xi)\,\phi(\xi)\,d\xi
%    \\&
    = \Du{\F\left(\PP_\theta f_{k_l}\right)}{\phi}
    \\&
     = \Du{\PP_\theta f_{k_l}}{\Fphi}
%    \\&
     \xrightarrow{l \to \infty} \Du{\PP_\theta f}{\Fphi}
     = \Du{\F\left(\PP_\theta f\right)}{\phi}.
  \end{split}
  \end{equation}
  Before using this result, note that by \equ{SKequ49},
  \begin{equation}
    \int_\Sm \Norm[L^2(\theta^\bot)]{|\xi|^{1/2} \left(\Ff(\xi) 
        - \widehat{f_{k_l}}(\xi)\right)}^2 \,d\theta
    \le |\Sm| \, \nxr{f - f_{k_l}}^2
    \xrightarrow{l \to \infty} 0,
  \end{equation}
  so there is again a subsequence $\left(f_{k_{l_m}}\right)_{m \in \N}$
  such that
  \begin{equation}
    \Norm[L^2(\theta^\bot)]{|\xi|^{1/2} 
      \left(\Ff(\xi) - \widehat{f_{k_{l_m}}}(\xi)\right)}
    \xrightarrow{m \to \infty} 0
  \end{equation}
  for almost all $\theta \in \Sm$.
  For such $\theta$,
%  \begin{equation}
%  \begin{split}
  \begin{align}
    \label{SKthm48cEstimate1}
    \int_{\theta^\bot \setminus B(0,1)} 
      \left| \Ff(\xi) - \widehat{f_{k_{l_m}}}(\xi) \right| \,|\phi(\xi)| \,d\xi
    &
    \le \int_{\theta^\bot \setminus B(0,1)} 
      |\xi|^{1/2} \left| \Ff(\xi) - \widehat{f_{k_{l_m}}}(\xi) \right| 
      \, |\phi(\xi)| \,d\xi
    \notag
    \\&
    \le \int_{\theta^\bot} 
      |\xi|^{1/2} \left| \Ff(\xi) - \widehat{f_{k_{l_m}}}(\xi) \right| 
      \, |\phi(\xi)| \,d\xi
    \notag
    \\&
    \le \Norm[L^2(\theta^\bot)]{|\xi|^{1/2} 
      \left( \Ff(\xi) - \widehat{f_{k_{l_m}}}(\xi) \right)} \,
      \Norm[L^2(\theta^\bot)]{\phi}
    \notag
    \\&
    \xrightarrow{m \to \infty} 0.
  \end{align}
%  \end{split}
%  \end{equation}
  Since $\phi(0)$ was assumed to be 0,
  \begin{equation}
    |\phi(\xi)| 
    = \left|\int_0^1 \nabla \phi(t \xi) \cdot \xi \, dt \right|
    \le \int_0^1 |\nabla \phi(t \xi)| \, |\xi| \, dt
    \le C_4 \, |\xi|,
  \end{equation}
  where $C_4 = \sqrt{\sum_{j=1}^n \Norm[0, e_j]{\phi}^2}$.
  Consequently,
  \begin{equation}
  \begin{split}
  \label{SKthm48cEstimate2}
    \int_{\theta^\bot \cap B(0,1)} &
      \left| \Ff(\xi) - \widehat{f_{k_{l_m}}}(\xi) \right| \,|\phi(\xi)| \,d\xi
    \\&
    \le C_4 \int_{\theta^\bot \cap B(0,1)} |\xi|^{1/2} \, |\xi|^{1/2} \, 
      \left| \Ff(\xi) - \widehat{f_{k_{l_m}}}(\xi) \right| \,d\xi
    \\&
    \le C_4 \Norm[L^2(\theta^\bot \cap B(0,1))]{|\xi|^{1/2}} \,
      \Norm[L^2(\theta^\bot \cap B(0,1))]{|\xi|^{1/2}
        \left( \Ff(\xi) - \widehat{f_{k_{l_m}}}(\xi) \right)}
    \\&
    \le C_5 \Norm[L^2(\theta^\bot)]{|\xi|^{1/2}
        \left( \Ff(\xi) - \widehat{f_{k_{l_m}}}(\xi) \right)}
    \xrightarrow{m \to \infty} 0.
  \end{split}
  \end{equation}
  Combining estimates \equ{SKthm48cEstimate1} and \equ{SKthm48cEstimate2}
  yields
  \begin{equation}
    \label{SKthm84cFfklmToFfInSsPrime}
    \left| \Du{\widehat{f_{k_{l_m}}}}{\phi} - \Du{\Ff}{\phi} \right|
    \le \int_{\theta^\bot}\left| \widehat{f_{k_{l_m}}}(\xi) - \Ff(\xi) \right|
    \, |\phi(\xi)| \, d\xi
    \xrightarrow{m \to \infty} 0.
  \end{equation}
  Equations \equ{SKthm38cIntFfklPsiToDuFPThetaFPsi} and
  \equ{SKthm84cFfklmToFfInSsPrime} together show that
  for almost all $\theta \in \Sm$,
  \begin{equation}
%  \begin{split}
    \Du{\F(\PP_\theta f)}{\phi}
%    &
%    = \Du{\PP_\theta f}{\Fphi}
%    \\&
%    = \lim_{m \to \infty} \Du{\PP_\theta f_{k_{l_m}}}{\Fphi}
%    \\&
%    = \lim_{m \to \infty} \Du{\F\left(\PP_\theta f_{k_{l_m}}\right)}{\phi}
%    \\&
    = \sqrt{2\pi} \, \lim_{m \to \infty} 
      \int_{\theta^\bot} \widehat{f_{k_{l_m}}}(\xi) \, \phi(\xi) \,d\xi
%    \\&
    = \sqrt{2\pi} \, \int_{\theta^\bot} \Ff(\xi) \, \phi(\xi) \,d\xi.
%  \end{split}
  \end{equation}
  This proves claim (c).

  In view of (d),
%  it suffices to show that
%  \begin{equation}
%    \int_\Sm \int_{B(0,R) \cap \thb} |\Ff(\xi)| \,d\xi \,d\theta
%  \end{equation}
%  is finite. In fact, 
  Lemma~\ref{lemma:SKlemma46} asserts that
  \begin{equation}
  \begin{split}
    \int_\Sm \int_{B(0,R) \cap \thb} |\Ff(\xi)| \,d\xi \,d\theta
    &
    = \int_\Sm \int_\thb 
      |\xi| \, |\xi|^{-1} \, |\Ff(\xi)| \, \chi_{B(0,R)}(\xi) \,d\xi \,d\theta
    \\&
    = |S^{n-2}| \int_\Rn |\xi|^{-1} \, |\Ff(\xi)| \, \chi_{B(0,R)}(\xi) \,d\xi
    \\&
    = |S^{n-2}| \int_{B(0,R)} |\xi|^{-1} \, |\Ff(\xi)| \,d\xi,
  \end{split}
  \end{equation}
  which is finite for any $R > 0$ by the assumption that 
  $|\xi|^{-1} \Ff(\xi) \in \Lloc(\Rn)$.
  Therefore $\Ff \in \Lloc(\thb)$ for almost all $\theta \in \Sm$.

  Fix $\theta \in \Sm$ such that $\Ff \in \Lloc(\thb)$,
  $\PP_\theta f \in \Ss'(\thb)$ and
  \begin{equation}
    \du{\F(\PP_\theta f)}{\phi}
    = \sqrt{2\pi} \int_\thb \Ff(\xi) \, \phi(\xi) \,d\xi
%   \int_\thb 
%     \left( \F\left(\PP_\theta f\right)(\xi) - \sqrt{2\pi} \,\Ff(\xi) \right) 
%     \phi(\xi) \,d\xi = 0
  \end{equation}
  whenever $\phi \in \Ss(\thb)$ and $\phi(0) = 0$.
  We have already proved that almost all points of $\Sm$ satisfy
  these conditions.

  Denote by $\delta_\thb$, $\F_\thb$ and $\F^{-1}_\thb$
  the delta distribution, the Fourier transform and 
  the inverse Fourier transform on $\thb$, respectively.
  By Lemma~\ref{lemma:DistributionsAgreeForPhiVanishingAt0},
  $\F_\thb(\PP_\theta f) = \sqrt{2\pi} \, \Ff + c \delta_\thb$
  for some constant $c \in \C$,
  so that 
  \begin{equation}
    \PP_\theta f = u + c,
    \qquad \text{where} \qquad
    u = \sqrt{2\pi} \, \F^{-1}_\thb(\Ff).
  \end{equation}
  Claim (d) will be proved once we succeed in showing that $c = 0$.

  Set $e = \frac{1}{|B^{n}|} \chi_{B(0,1)}$ and $e_\rho(x) = \rho^{-n} e(x/\rho)$.
  Lemmata \ref{lemma:SKthm42} and \ref{lemma:SKlemma314}(b) show that
  \begin{equation}
    \int_\Sm \Du{\PP_\theta f}{\PP_\theta e_\rho} \,d\theta
    = \Du{R_1 * f}{e_\rho} 
    \xrightarrow{\rho \to \infty} 0.
  \end{equation}
  Therefore the sequence $(e_\rho)_{\rho \in \Z_+}$ 
  has a subsequence $(e_{\rho_k})_{k \in \N}$ such that
  \begin{equation}
    \Du{\PP_\theta f}{\PP_\theta e_{\rho_k}} 
    \xrightarrow{k \to \infty} 0
    \qquad \text{for almost all } \theta \in \Sm.
  \end{equation}
  Since
  \begin{equation}
  \begin{split}
    c
    &
    = \int_\Rn c \, e_{\rho_k}(x) \,dx
    \\&
    = \int_\thb c \infint \, e_{\rho_k}(x' + t\theta) \,dt \,dx'
    \\&
%    = \int_\thb c \, \PP_\theta e_{\rho_k}(x') \,dx'
%    \\&
    = \Du{c}{\PP_\theta e_{\rho_k}}
    \\&
    = \Du{\PP_\theta f}{\PP_\theta e_{\rho_k}} - \Du{u}{\PP_\theta e_{\rho_k}},
  \end{split}
  \end{equation}
  it will suffice to show that the second term on the right hand side
  approaches zero as $k \to \infty$. 
  We shall do this with the help of Lemma~\ref{lemma:SKlemma314}(a)
  in $\thb \cong \R^{n-1}$, with $\PP_\theta e$ in place of $e$.

  The change of variable $s = t/\rho$, $ds = dt/\rho$ shows that
  \begin{equation}
%  \begin{split}
    \PP_\theta e_\rho(x) 
%    &
    = \infint \rho^{-n} \, e\left(\frac{x + t\theta}{\rho}\right) \,dt
%    \\&
    = \rho^{1-n} \infint e\left( \frac{x}{\rho} + s\theta \right) \,ds
%    \\&
%    = \rho^{1-n} \, \PP_\theta e\left(\frac{x}{\rho}\right)
%    \\&
    = \left(\PP_\theta e\right)_\rho(x)
%  \end{split}
  \end{equation}
  since $\PP_\theta e$ is a function on the $n-1$-dimensional space $\thb$.
  The function $\PP_\theta e$ is in $L^\infty_0(\thb)$, since
  \begin{equation}
    \PP_\theta e(x) 
    = \frac{1}{|B^{n}|}\infint \chi_{B(0,1)} (x + t\theta) \,dt
    = \frac{2}{|B^{n}|} \max\left(0, \sqrt{1 - |x|^2}\right).
  \end{equation}
  The function $\Fu$ can be written as 
  $\Fu = \Fu \chi_{B(0,1)} + \Fu \chi_{\Rn \setminus B(0,1)}$,
  where
  \begin{equation}
    \Fu \chi_{B(0,1)} 
    = \sqrt{2\pi} \, \Ff \, \chi_{B(0,1)}
    \in L^1(\thb),
  \end{equation}
  as $\Ff \in \Lloc(\thb)$ by our choice of $\theta$, and
  \begin{equation}
    \Fu \chi_{\Rn \setminus B(0,1)} \in L^2(\thb)
    \qquad \text{for almost all }\theta \in \Sm,
  \end{equation}
  which can be seen as follows.
  Since
  \begin{equation}
  \begin{split}
    \int_\Sm \int_{\thb \setminus B(0,1)} |\Ff(\xi)|^2 \,d\xi \,d\theta
    &
    \le 
    \int_\Sm \int_{\thb \setminus B(0,1)} |\xi| \, |\Ff(\xi)|^2 \,d\xi\,d\theta
    \\&
    \le \int_\Sm \int_\thb |\xi| \, |\Ff(\xi)|^2 \,d\xi\,d\theta
    \\&
    = |S^{n-2}| \, \nl{\Ff}{2}^2
    = |S^{n-2}| \, \nl{f}{2}^2
    < \infty
  \end{split}
  \end{equation}
  by Lemma~\ref{lemma:SKlemma46},
  \begin{equation}
    \norm[L^2(\thb)]{\Fu \chi_{\Rn \setminus B(0,1)}}
%    = \sqrt{2\pi} \, \norm[L^2(\thb)]{\Ff \chi_{\Rn \setminus B(0,1)}}
    = \sqrt{2\pi} \, 
      \int_\thb |\Ff(\xi)|^2 \chi_{\Rn \setminus B(0,1)}(\xi) \,d\xi < \infty
  \end{equation}
  for almost all $\theta \in \Sm$.

  All assumptions of Lemma~\ref{lemma:SKlemma314}(a) are therefore satisfied,
  whence we see that
  \begin{equation}
    \du{u}{\PP_\theta e_\rho}
    = \du{u}{(\PP_\theta e)_\rho} 
    \xrightarrow{\rho \to \infty} 0.
  \end{equation}
  This completes the proof.
\end{tod}


The following theorem finally gives 
the approximate parallel beam reconstruction formula.

\begin{theorem}
  \label{thm:SKthm51}
  Let $n \ge 2$ and $e \in \Dxr(\Rn) \cap H^{1/2}(\Rn)$ be such that
  $|\xi|^{-1} \Fe(\xi) \in \Lloc(\Rn)$.
  Then for almost all $\theta \in \Sm$,
  \begin{enumerate}
  \vspace{-2ex}
  \item[(a)] 
    $\PP_\theta e \in \Lloc(\thb)$
  \item[(b)] with any coordinates on $\thb$,
    $D_j (\PP_\theta e) \in L^2(\thb)$ and
    \begin{equation}
      \F\big( D_j (\PP_\theta e) \big)(\xi) 
      = \sqrt{2\pi} \, i \, \xi_j \, \Fe(\xi)
    \end{equation}
    for almost all $\xi \in \thb$
  \item[(c)] the function
    \begin{equation}
      \Lambda \PP_\theta e 
      = - \sum_{j=1}^{n-1} \pv\dodo{R_1}{x_j} * D_j \left(\PP_\theta e \right),
    \end{equation}
%    where $R_1(x) = b_{n-1} \, |x|^{2-n}$ is the Riesz kernel 
    where $\pv \dodo{R_1}{x_j} *$ is the Riesz transform
    on $\thb \cong \R^{n-1}$, is in $L^2(\thb)$ and 
    \begin{equation}
      \F(\Lambda \PP_\theta e)(\xi) = \sqrt{2\pi} \, |\xi| \, \Fe(\xi)
    \end{equation}
    almost everywhere
  \item[(d)] if $f \in L^2_0(\Rn)$, then for all $x \in \Rn$
    \begin{equation}
      \label{SKthm51d}
      e * f(x) = \int_\Sm k * \PP_\theta f(E_\theta x) \,d\theta,
    \end{equation}
    where $k = \frac{b_n}{2} \Lambda \PP_\theta e$.
  \end{enumerate}
\end{theorem}

\begin{tod}
  Claim (a) follows immediately from Theorem~\ref{thm:SKthm48}(b).
%  since
%  Theorem~\ref{thm:SKthm48}(b) implies that $\PP_\theta e \in \Lloc(\thb)$,
%  and $D_j(\PP_\theta e) \in L^2(\thb) \subset L^2_{\textup{loc}}(\thb)$.
  In view of claim (b),
  Lemma~\ref{lemma:SKlemma46} asserts that
  \begin{equation}
  \begin{split}
    \int_\Sm \int_\thb |\xi|^2 \, |\Fe(\xi)|^2 \,d\xi \,d\theta
    &
    = |S^{n-2}| \int_\Rn |\xi| \, |\Fe(\xi)|^2 \,d\xi
    \\&
    \le |S^{n-2}| \int_\Rn (1 + |\xi|^2)^{1/2} |\Fe(\xi)|^2 \,d\xi
    \\&
    = |S^{n-2}| \, \norm[H^{1/2}(\Rn)]{e}
    < \infty.
  \end{split}
  \end{equation}
  Therefore almost all $\theta \in \Sm$ satisfy
  \begin{equation}
    \label{SKthm51xiFeInL2Thb}
    \norm[L^2(\thb)]{|\xi| \, \Fe(\xi)}^2
    = \int_\thb |\xi|^2\, |\Fe(\xi)|^2 \,d\xi
    < \infty
  \end{equation}
  and the claims of Theorem~\ref{thm:SKthm48}. Fix such a $\theta$.
  By Theorem~\ref{thm:SKthm48}(d), 
  $\F(\PP_\theta e) = \sqrt{2\pi} \, \Fe$ almost everywhere on $\thb$, 
  and therefore 
  $\F(D_j(\PP_\theta e))(\xi) = \sqrt{2\pi} \, i \, \xi_j \, \Fe(\xi)$,
  which is in $L^2(\thb)$ because
  \begin{equation}
    \norm[L^2(\thb)]{D_j(\PP_\theta e)}
    = \sqrt{2\pi} \, \norm[L^2(\thb)]{\xi_j \, \Fe(\xi)}
    \le \sqrt{2\pi} \, \norm[L^2(\thb)]{|\xi| \, \Fe(\xi)}
    < \infty.
  \end{equation}
  This proves (b). 

  Since $D_j(\PP_\theta e) \in L^2(\thb)$, 
  Theorem~\ref{thm:FourierTransformOfRieszTransform} in $\thb \cong \R^{n-1}$
  shows that
  \begin{equation}
    \Lambda \PP_\theta e
    = - \sum_{j=1}^{n-1} \pv \dodo{R_1}{x_j} * D_j \PP_\theta e 
    \in L^2(\thb),
  \end{equation}
  and that
  \begin{equation}
    \F\left(\Lambda \PP_\theta e\right)(\xi)
%    = - \sum_{j=1}^{n-1} \F\left(\pv \dodo{R_1}{x_j} * D_j \PP_\theta e\right)
    = - \sum_{j=1}^{n-1} i \, 
      \frac{\xi_j}{|\xi|} \, i \, \xi_j \, \F(\PP_\theta e)(\xi)
    = \sqrt{2\pi} \, \Fe(\xi) \, |\xi|^{-1} \sum_{j=1}^{n-1} \xi_j^2
    = \sqrt{2\pi} \, |\xi| \, \Fe(\xi),
  \end{equation}
  as claimed in (c).

  Claim (d) will be proved through a usual limit argument:
  we shall first derive the result for test functions,
  and then show that when more general functions are approximated 
  by test functions, 
  the result holds at the limit also for the more general functions.
  
  Assume that $\phi \in L^2_0(\Rn)$ and $\supp \phi \subset B(0,R)$; 
  we shall soon apply the results that we are going to derive, 
  with $\phi$ a function closely related to $f$.
  Then
  \begin{equation}
    \norm[L^2(\thb)]{\PP_\theta \phi}^2
    = \int_\thb \Big| \infint \phi(y + t\theta) \,dt \, \Big|^2 \,dy,
  \end{equation}
  and applying the triangle inequality and Hölder's inequality 
  to the inner integral gives
  \begin{equation}
  \begin{split}
    \label{SKthm51suluissa2}
    \norm[L^2(\thb)]{\PP_\theta \phi}^2
    &
    \le \int_\thb \infint |\phi(y + t\theta)|^2 
      \, \chi_{(-R,R)}(y + t\theta)^2 \,dt \,dy
    \\&
    \le \int_\thb 2 R \infint |\phi(y + t\theta)|^2 \,dt \,dy
    \\&
    = 2 R \, \norm[L^2(\Rn)]{\phi}^2
    < \infty,
  \end{split}
  \end{equation}
  so that $\PP_\theta \phi \in L^2(\thb)$.
  Also, $\supp \PP_\theta \phi \subset B(0,R)$, since
  whenever $y \in \thb \setminus B(0,R)$,
  $|y + t\theta| = \sqrt{|y|^2 + t^2} \ge R$ for all $t$.

  Choose a sequence of test functions $\phi_k \in \DD(\Rn)$, such that
  $\supp \phi_k \subset B(0,R)$ and $\norm[L^2(\Rn)]{\phi_k - \phi} \to 0$.
  Then
  \begin{equation}
  \begin{split}
    \int_\thb \Lambda \PP_\theta e(y) \, \PP_\theta \phi_k(y) \,dy
    &
    = \Du{\Lambda \PP_\theta e}{\PP_\theta \phi_k}
    \\&
    = \Du{\F \Lambda \PP_\theta e}{\Fi \Fi \F \PP_\theta \phi_k}.
  \end{split}
  \end{equation}
  Now note that by Lemma~\ref{lemma:ProjectionSliceTheorem}, 
  $\F \PP_\theta \phi_k (\xi) = \sqrt{2\pi} \, \widehat{\phi_k}(\xi)$
  and by \equ{FourierTransformAntiInvolutivity}, $\Fi\Fi = \refl{\cdot}$.
  By claim (c) just proved above, 
  $\F \Lambda \PP_\theta e (\xi) = \sqrt{2\pi} \, |\xi| \, \Fe(\xi)$.
  Therefore,
  \begin{equation}
    \int_\thb \Lambda \PP_\theta e(y) \, \PP_\theta \phi_k(y) \,dy
    = 2 \pi \int_\thb |\xi| \, \Fe(\xi) \, \widehat{\phi_k}(-\xi) \, d\xi,
  \end{equation}
  and consequently
  \begin{equation}
    \int_\Sm \int_\thb \Lambda \PP_\theta e(y) \, \PP_\theta \phi_k(y) 
    \,dy \,d\theta
    = 2\pi \int_\Sm \int_\thb |\xi| \,\Fe(\xi) \,\widehat{\phi_k}(-\xi) 
      \,d\xi \,d\theta.
  \end{equation}
  As $\norm[L^1(\thb)]{|\xi| \, \Fe(\xi) \,\widehat{\phi_k}(-\xi)} \le
  \norm[L^2(\thb)]{|\xi| \, \Fe(\xi)} \norm[L^2(\thb)]{\widehat{\phi_k}(-\xi)}
  < \infty$
  for almost all $\theta \in \Sm$ by \equ{SKthm51xiFeInL2Thb}, 
  Lemma~\ref{lemma:SKlemma46} can be applied 
  separately to the positive and negative parts of 
  $\Fe(\xi) \widehat{\phi_k}(-\xi)$, yielding
  \begin{equation}
  \begin{split}
    \int_\Sm \int_\thb \Lambda \PP_\theta e(y) \, \PP_\theta \phi_k(y) 
    \,dy \,d\theta
    &
    = 2\pi \, |S^{n-2}| \int_\Rn \Fe(\xi) \, \widehat{\phi_k}(-\xi) \,d\xi
    \\&
    = 2\pi \, |S^{n-2}| \du{\F e}{\Fi \Fi \F \phi_k}
    \\&
    = 2\pi \, |S^{n-2}| \du{e}{\phi_k}
    \\&
    = 2\pi \, |S^{n-2}| \int_\Rn e(y) \, \phi_k(y) \,dy.
  \end{split}
  \end{equation}
  The same holds at the limit for $\phi$, which can be seen as follows.
  By Hölder's inequality and \equ{SKthm51suluissa2},
  \begin{equation}
  \begin{split}
    \Big| \int_\Sm \int_\thb \Lambda \PP_\theta & e(y) \, \big(
        \PP_\theta \phi_k(y) - \PP_\theta \phi(y) \big) \,dy\,d\theta \, \Big|
    \\&
    \le \int_\Sm \norm[L^2(\thb)]{\Lambda \PP_\theta e}
      \norm[L^2(\thb)]{\PP_\theta \phi_k - \PP_\theta \phi} \,d\theta
    \\&
    \le |\Sm| \norm[L^2(\thb)]{\Lambda \PP_\theta e} \,
      \sqrt{2R} \, \norm[L^2(\Rn)]{\phi_k - \phi}
    \xrightarrow{k \to \infty} 0.
    \end{split}
  \end{equation}
  Since
  \begin{equation}
    \Big| \int_\Rn e(y) \big(\phi_k(y) - \phi(y)\big) \,dy \, \Big|
    \le \norm[L^2(\Rn)]{e} \norm[L^2(\Rn)]{\phi_k - \phi}
%    \le \norm[\Dxr(\Rn)]{e} \norm[L^2(\Rn)]{\phi_k - \phi}
    \xrightarrow{k \to \infty} 0,
  \end{equation}
  this implies that
  \begin{equation}
  \begin{split}
    2\pi \, |S^{n-2}| \int_\Rn e(y) \, \phi(y) \,dy
    &
    = \lim_{k \to \infty} 2\pi \, |S^{n-2}| \int_\Rn e(y) \, \phi_k(y) \,dy
    \\&
    = \lim_{k \to \infty} \int_\Sm \int_\thb 
      \Lambda \PP_\theta e(y) \, \PP_\theta \phi_k(y) \,dy \,d\theta
    \\&
    = \int_\Sm \int_\thb \Lambda \PP_\theta e(y) \, \PP_\theta \phi(y) \,dy
      \,d\theta.
  \end{split}
  \end{equation}
  If we now choose $\phi(y) = f(x - y)$ with $x \in \Rn$, 
  we observe that using $\tau = t - x \cdot \theta$, $d\tau = - dt$,
  \begin{equation}
  \begin{split}
    \PP_\theta \phi(y)
    &
    = \infint f(x - y - t\theta) \,dt
    \\&
    = \infint f(E_\theta x + (x \cdot \theta) \, \theta - y - t\theta) \,dt
    \\&
    = - \int_{\infty}^{-\infty} f(E_\theta x - y + \tau \theta) \,d\tau
    \\&
    = \PP_\theta f(E_\theta x - y).
  \end{split}
  \end{equation}
  Therefore,
  \begin{equation}
  \begin{split}
    e * f(x)
    &
    = \int_\Rn e(y) \, f(x - y) \, dy
    \\&
    = \frac{1}{2\pi \, |S^{n-2}|}
      \int_\Sm \int_\thb \Lambda \PP_\theta e(y) \,\PP_\theta f(E_\theta x - y)
      \,dy \,d\theta
    \\&
    = \frac{b_n}{2}
      \int_\Sm \left(\Lambda \PP_\theta e * \PP_\theta f\right)(E_\theta x) 
      \,d\theta,
  \end{split}
  \end{equation}
  as claimed.
\end{tod}




\section{Reconstruction Formulae}
\label{sec:ReconstructionFormulae}


The different reconstruction formulae can now be summarized as follows.
The exact reconstruction formula is an immediate consequence of 
Theorem~\ref{thm:FourierTransformOfRiestPotentialInDxr}:

\begin{theorem}
  \label{thm:ExactReconstruction}
  If $n \in \Z_+$, 
  $f \in \Dxr(\Rn)$ and $|\xi|^{-1} \Ff(\xi) \in \Lloc(\Rn)$,
  then for almost all $x$,
  \begin{equation}
    \Lambda(R_1 * f)(x) = f(x).
  \end{equation}
\end{theorem}

\begin{tod}
%  For the tempered distribution $R_1 * f$,
%%  Because $L^{\infty} \subset \Ss'$, $R_1 * f$ is a tempered distribution
%%  and we can calculate
%  \begin{equation}\nt
%    \F(D_j (R_1*f)) = i \xi_j \F(R_1*f) = i \xi_j \,|\xi|^{-1} \, \Ff
%    = \F \left(\pv \dodo{R_1}{x_j} * f \right).
%  \ent\end{equation}
%  Therefore the Parseval formula~\equ{PFormulaNorm}
%  and Corollary~\ref{cor:CalderonZygmundSummary} show that
%  $\nl{D_j (R_1*f)}{2} = \nl{\pv \dodo{R_1}{x_j} * f}{2} \le M \, \nl{f}{2}
%  < \infty$.
%  Hence,
%  \begin{equation}\nt
%    \F(\Lambda(R_1*f))
%    = - \sum_{j=1}^n i\, \frac{\xi_j}{|\xi|}\, i\, \frac{\xi_j}{|\xi|}\, \Ff
%    = \Ff,
%  \ent\end{equation}
%  which implies that $\Lambda(R_1*f) = f$ in the $L^2$ sense and consequently
%  also almost everywhere.
  By Theorem~\ref{thm:FourierTransformOfRiestPotentialInDxr},
  \begin{equation}
    \F(\Lambda (R_1 * f))(\xi)
    = |\xi| \, |\xi|^{-1} \, \Ff(\xi)
    = \Ff(\xi).
  \end{equation}
  Therefore $\Lambda(R_1 * f) = f$ as distributions.
  Since $f \in \Dxr \subset \Lloc$, this also implies 
  equality almost everywhere.
\end{tod}

As $\Lambda$ is a continuous operator from $H^1$ to $L^2$
by Theorem~\ref{thm:LambdaAspvConvolution},
the estimate for $\nl{\Lambda f}{2}$
depends on the $L^2$-norms of the derivatives of $f$ through $\norm[H^1]{f}$.
This is about as good as we can get, because 
\begin{equation}
  \Lambda = - \sum_{j=1}^n \pv \dodo{R_1}{x_j} * D_j
\end{equation}
explicitly involves the derivatives, 
and the Riesz transform $\pv \dodo{R_1}{x_j} *$ is not a smoothing operator.
The exact reconstruction is, therefore, sensitive to noise
in the measurements.


The approximate reconstruction formula is obtained from 
Theorem~\ref{thm:SKthm51}:

\begin{corollary}
  \label{cor:ApproxReconstruction}
  Let $n \ge 2$ and $e \in \Dxr(\Rn) \cap H^{1/2}(\Rn)$ be such that 
  $|\xi|^{-1} \Fe(\xi) \in \Lloc(\Rn)$. 
  If $f \in L^2_0(\Rn)$ and $\supp f \subset B(0,R)$, then
  for all $x \in \Rn$, 
  \begin{equation}
  \begin{split}
    \label{ApproxReconstructionFormula}
    e * & f (x)
    = \frac{b_n}{2} \int_\Sm \int_{\theta^\bot} 
      \Lambda \PP_\theta e (E_\theta x - y) \, \PP_\theta f (y) \,dy \,d\theta
    \\&
    = \frac{b_n}{4R} \int_A \int_\Sm
      \Lambda \PP_\theta e \bigl(E_\theta (x - a)\bigr) \,
      \bigl( \DD_a f (\theta) + \DD_a f (- \theta) \bigr) \,
      |a \cdot \theta| \,d\theta \,da,
  \end{split}
  \end{equation}
  where $A = S^{n-1}(0,R)$.
\end{corollary}

\begin{tod}
  The first equality is Equation~\equ{SKthm51d} 
  of Theorem~\ref{thm:SKthm51}(d).
  To obtain the second, make the substitution 
  $y = E_\theta a$, $dy = \frac{|a \cdot \theta|}{R} \, da$,
  where $y$ runs twice over
  $B(0,R) \cap \theta^\bot$ (containing $\supp \PP_\theta f$)
  as $a$ runs over the two hemispheres of $A$ 
  (see Figure~\ref{fig:subst1}). This gives
  \begin{equation}\nt
  \begin{split}
    \label{ApproxDivBeamReconstructionFormula}
    e * & f (x) 
    = \frac{b_n}{4R} \int_\Sm \int_A 
      \bigl(\Lambda \PP_\theta e (E_\theta (x - a)) \bigr) \, 
      \PP_\theta f (E_\theta a) \, |a \cdot \theta| \,da \,d\theta
    \\ &
    = \frac{b_n}{4R} \int_A \int_\Sm
      \bigl( \Lambda \PP_\theta e (E_\theta (x - a)) \bigr) \,
      \bigl( \DD_a f (\theta) + \DD_a f (- \theta) \bigr) \,
      |a \cdot \theta| \,d\theta \,da.
  \end{split}
  \ent\end{equation}
  To justify changing the order of integration by Fubini's theorem, 
  note that using Hölder's inequality twice, first on the outer integral
  and then on the inner one, yields
  \begin{equation}
  \begin{split}
    \bigg(\int_\Sm \int_\thb | & \Lambda \PP_\theta e(E_\theta x - y)| \,
      |\PP_\theta f(y)| \, dy \, d\theta \bigg)^2
    \\&
    \le 
    \int_\Sm \Big(\int_\thb |\Lambda \PP_\theta e(E_\theta x - y)| 
             |\PP_\theta f(y)| \,dy \Big)^2 d\theta
    \int_\Sm 1^2 \, d\theta
    \\&
    \le 
    |\Sm| \int_\Sm \int_\thb |\Lambda \PP_\theta e(E_\theta x - y)|^2 \,dy
                   \int_\thb |\PP_\theta f(y)|^2 \,dy \, d\theta.
  \end{split}
  \end{equation}
  This can be seen to be finite by analysing both inner integrals separately.
  Since
  \begin{equation}
  \begin{split}
    \int_\thb |\Lambda \PP_\theta e(E_\theta x - y)|^2 \,dy
    &
    = \int_\thb |\Lambda \PP_\theta e(y)|^2 \,dy
    \\&
    = \norm[L^2(\thb)]{\Lambda \PP_\theta e}
%    \\&
    = \norm[L^2(\thb)]{\F(\Lambda\PP_\theta e)},
  \end{split}
  \end{equation}
  Theorem~\ref{thm:SKthm51}(c) and Lemma~\ref{lemma:SKlemma46}
  assert that
  \begin{equation}
  \begin{split}
    \int_\thb |\Lambda \PP_\theta e(E_\theta x - y)|^2 \,dy
    &
    = \int_\thb |\xi|^2 |\Fe(\xi)|^2 \,d\xi
    \\&
    = |S^{n-2}| \int_\Rn |\xi| |\Fe(\xi)|^2 \,d\xi
    \\&
    \le |S^{n-2}| \, \norm[H^{1/2}(\Rn)]{e}^2 
    =: C_1.
  \end{split}
  \end{equation}
  If $\supp f \subset B(0,\tilde{R})$, then \equ{SKthm51suluissa2}
  tells that
  \begin{equation}
%  \begin{split}
    \int_\thb |\PP_\theta f(y)|^2 \,dy
%    &
    = \norm[L^2(\thb)]{\PP_\theta f}^2
%    \\&
    \le 2 \tilde{R} \, \norm[L^2(\Rn)]{f}^2
    =: C_2.
%  \end{split}
  \end{equation}
  Consequently,
  \begin{equation}
    \bigg(\int_\Sm \int_\thb | \Lambda \PP_\theta e(E_\theta x - y)| \,
      |\PP_\theta f(y)| \, dy \, d\theta \bigg)^2
    \le |\Sm|^2 \, C_1 \, C_2
    < \infty.
  \end{equation}
\end{tod}

The reconstruction formula for $\Lambda e * f$ is obtained 
by replacing $e$ by $\Lambda e$ above:

\begin{theorem}
  \label{thm:EConvLambdaFReconstruction}
  Let $n \ge 2$ and 
  let $e \in H^{3/2}(\Rn) \cap \Dxr(\Rn)$ 
  be such that $\Lambda e$, $D_1 e$, \ldots, $D_n e \in \Dxr(\Rn)$.
  If $f \in L^2_0(\Rn)$, then
  \begin{equation}
  \label{EConvLambdaFReconstructionFormula} 
  \begin{split}
    \Lambda(e * & f)(x) = (\Lambda e) * f(x)
\\
    = & - \frac{b_n}{2} \int_\Sm 
      \big(\laplace \PP_\theta e\big) * \big(\PP_\theta f\big)(E_\theta x) 
      \,d\theta
\\
    = & - \frac{b_n}{4R} \int_A \int_\Sm 
      \laplace \PP_\theta e(E_\theta (x - a))
      \left( \DD_a f(\theta) + \DD_a f(-\theta) \right) |a \cdot \theta|
      \,d\theta \, da
  \end{split}
  \end{equation}
  for all $x \in \Rn$.
  If, in addition, $e$ has compact support, then
  \begin{equation}
    \Lambda(e * f) = e * \Lambda f.
  \end{equation}
\end{theorem}

\begin{tod}
  First observe that
  \begin{equation}
    \label{EConvLambdaFReconstruction:LambdaEInH1Over2}
    \norm[H^{1/2}]{\Lambda e}
    = \nl{(1 + |\xi|^2)^{1/4} |\xi| \, \Fe(\xi)}{2}
    \le \nl{(1 + |\xi|^2)^{3/4} \Fe(\xi)}{2}
    = \norm[H^{3/2}]{e} < \infty,
  \end{equation}
  and that
  \begin{equation}
  \begin{split}
    \label{EConvLambdaFReconstruction:FLambdaEInL1Loc}
    \int_K |\xi|^{-1} \, |\F \Lambda e(\xi)| \,d\xi
    &
    = \int_K |\Fe(\xi)| \,d\xi
    \\&
    \le \norm[L^2]{\Fe} \norm[L^2]{\chi_K}
    \\&
    \le \norm[H^{3/2}]{e}\, \sqrt{m(K)}
    < \infty
  \end{split}
  \end{equation}
  whenever $K$ is compact.
  Therefore Corollary~\ref{cor:ApproxReconstruction}
  can be applied to $\Lambda e$:
  \begin{align}
    \label{EConvLambdaFReconstructionAlmostThere}
    (\Lambda e) * f (x)
    &
    = \frac{b_n}{2} \int_\Sm 
      \big(\Lambda \PP_\theta \Lambda e\big)*\big(\PP_\theta f\big)(E_\theta x)
      \, d\theta
    \\&
    \notag
    = \frac{b_n}{2R} \int_A \int_\Sm 
      \big(\Lambda \PP_\theta \Lambda e(E_\theta(x - a))\big) 
      \big(\DD_a f(\theta) + \DD_a f(-\theta)\big) \, |a \cdot \theta| 
      \, d\theta \, da.
  \end{align}
  Now Theorem~\ref{thm:SKthm51}(c), applied to $\Lambda e$, 
  asserts that for almost all $\theta \in \Sm$,
  \begin{equation}
    \F(\Lambda \PP_\theta \Lambda e)
    = \sqrt{2\pi} \, |\xi| \, \F(\Lambda e)
    = \sqrt{2\pi} \, |\xi|^2 \, \Fe(\xi)
    = - \sum_{j=1}^n \sqrt{2\pi} \, i \xi_j \,\F(D_j e).
  \end{equation}
  Since
  \begin{equation}
    |\F{D_j e}|
    = |\xi_j| \, \Fe(\xi)
    \le |\xi| \, \Fe(\xi)
    = |\F \Lambda e|,
  \end{equation}
  Equations \equ{EConvLambdaFReconstruction:LambdaEInH1Over2} and
  \equ{EConvLambdaFReconstruction:FLambdaEInL1Loc} show that
  $D_j e \in H^{1/2}$ and $|\xi|^{-1} \F(D_j e) \in \Lloc$.
  Therefore Theorem~\ref{thm:SKthm51}(b) can be applied to $D_j e$, 
  yielding that
  \begin{equation}
    \sqrt{2\pi} \, i \xi_j \, \F(D_j e)
    = \F(D_j \PP_\theta D_j e)
  \end{equation}
  for almost all $\theta \in \Sm$.
  Now $\PP_\theta D_j e = D_j \PP_\theta e$ as distributions
  for almost all $\theta \in \Sm$, 
  which can be seen as follows. 
  Choose any $\phi \in \DD(\thb)$.
  By Theorem~\ref{thm:SKthm48}(b), 
  $\PP_\theta e \in \Lloc(\thb)$ for almost all $\theta \in \Sm$, 
  so that
  \begin{equation}
    \du{D_j \PP_\theta e}{\phi}
    = - \Du{\PP_\theta e}{\dodo{\phi}{x_j}}
    = - \int_{\supp \phi} \infint e(x + t\theta) \, dt 
      \, \dodo{\phi}{x_j}(x) \,dx.
  \end{equation}
  Using Fubini's theorem, partial integration and the fact
  that $D_j e \in H^{1/2} \subset \Lloc$, gives
  \begin{equation}
  \begin{split}
    \du{D_j \PP_\theta e}{\phi}
    &
    = - \infint \int_{\supp \phi} e(x + t\theta) \,\dodo{\phi}{x_j}(x) \,dx\,dt
    \\&
    = \infint \int_{\supp \phi} D_j e(x + t\theta) \, \phi(x) \,dx \,dt
    \\&
    = \int_{\supp\phi} \infint D_j e(x + t\theta) \,dt \, \phi(x) \,dx
    \\&
    = \du{\PP_\theta D_j e}{\phi}
  \end{split}
  \end{equation}
  and shows that $\PP_\theta D_j e = D_j \PP_\theta e$ indeed,
  and consequently
  \begin{equation}
    \F(\Lambda \PP_\theta \Lambda e)
    = - \sum_{j=1}^n \F(D_j D_j \PP_\theta e)
    = \F(- \laplace \PP_\theta e).
  \end{equation}
  Therefore $\Lambda \PP_\theta \Lambda e = - \laplace \PP_\theta e$,
  and substituting this into \equ{EConvLambdaFReconstructionAlmostThere}
  proves the first part of the claim. 
  Since $e$, $\Lambda e \in L^2 \subset \Ss'$ and 
  $f \in L^2_0 \subset \DD'_0$,
  \begin{equation}
    \F\big(\Lambda (e * f)\big)(\xi)
    = |\xi| \, \Fp \, \Fe(\xi) \Ff(\xi)
    = \F\big((\Lambda e) * f)\big)(\xi).
  \end{equation}
  If $e$ has bounded support, then $e \in \DD'_0$ and we can write
  \begin{equation}
    \F\big(\Lambda (e * f)\big)(\xi)
    = \F(e * \Lambda f)(\xi).
  \end{equation}
\end{tod}



\section{Stability Results}
\label{sec:Stability}

In contrast with reconstructing $f = \Lambda \, R_1 * f$,
which is sensitive to noise in the measurements,
reconstructing $e * f$ is a stable operation.
The following theorem tells that the error in $e * f$ is uniformly
bounded by the $L^2$ norm of the error in the measurements $\DD f$:

\begin{theorem}
  \label{thm:ApproxReconstructionStability}
  If $n \ge 2$ and $e \in \Dxr(\Rn) \cap H^{1/2}(\Rn)$ is such that 
  $|\xi|^{-1} \Fe(\xi) \in \Lloc(\Rn)$, 
  then there exists a constant $C > 0$ such that
  \begin{equation}
    |e * f(x)| 
    \le C \, \norm[L^2(\Sm(0, R) \times \Sm)]{\DD f}
  \end{equation}
  for all $f \in L^2_0(\Rn)$ with $\supp f \subset B(0,R)$ and all $x \in \Rn$.
\end{theorem}

\begin{tod}
  By Corollary~\ref{cor:ApproxReconstruction},
  Young's inequality on $\thb$ and Hölder's inequality on $\Sm$,
  \begin{equation}
  \begin{split}
    \label{ApproxContinuityFirstEstimate}
    |e * f(x)|^2 
    &
    \le \frac{b_n^2}{4} \left[ 
      \int_\Sm \big|\Lambda \PP_\theta e * \PP_\theta f (E_\theta x) \big|
      \,d\theta \right]^2
    \\&
    \le \frac{b_n^2}{4} \left[
      \int_\Sm \norm[L^2(\thb)]{\Lambda \PP_\theta e}
               \norm[L^2(\thb)]{\PP_\theta f}
      \,d\theta \right]^2
    \\&
    \le \frac{b_n^2}{4}
      \int_\Sm \norm[L^2(\thb)]{\Lambda \PP_\theta e}^2 \, d\theta
      \int_\Sm \norm[L^2(\thb)]{\PP_\theta f}^2 \, d\theta.
  \end{split}
  \end{equation}
  Theorem~\ref{thm:SKthm51}(c) asserts that 
  \begin{equation}
    \norm[L^2(\thb)]{\Lambda \PP_\theta e}^2
    = 2\pi \, \norm[L^2(\thb)]{|\xi| \, \Fe(\xi)}^2
    = 2\pi \, \int_\thb |\xi|^2 \, |\Fe(\xi)|^2 \,d\xi,
  \end{equation}
  whence 
  \begin{equation}
  \begin{split}
    \int_\Sm \norm[L^2(\thb)]{\Lambda \PP_\theta e}^2 \,d\theta
    &
    = 2\pi \int_\Sm \int_\thb |\xi|^2 \, |\Fe(\xi)|^2 \,d\xi \,d\theta
    \\&
    = 2\pi \, |S^{n-2}| \int_\Rn |\xi| \, |\Fe(\xi)|^2 \,d\xi
    = 2\pi \, |S^{n-2}| \, \norm[H^{1/2}]{e}^2
  \end{split}
  \end{equation}
  by Lemma~\ref{lemma:SKlemma46}. 
  Performing the already familiar change of coordinates 
  $y = E_\theta a$, $dy = \frac{|a \cdot \theta|}{R} \, da$,
  where $y$ runs twice over
  $B(0,R) \cap \theta^\bot$ (containing $\supp \PP_\theta f$)
  as $a$ runs over the two hemispheres of $A := \Sm(0, R)$
  (see Figure~\ref{fig:subst1}),
  shows that the latter integral in \equ{ApproxContinuityFirstEstimate}
  is equal to
  \begin{equation}
  \begin{split}
    \int_\Sm \int_\thb |\PP_\theta f(y)|^2 \,dy \,d\theta
    &
    = \frac{1}{2} \int_\Sm \int_A |\PP_\theta f(E_\theta a)|^2 
      \, \frac{|a \cdot \theta|}{R} \, da \, d\theta
    \\&
    \le \int_\Sm \int_A |\DD_a f(\theta)|^2 \,da \,d\theta
    = \norm[L^2(A \times \Sm)]{\DD f}^2.
  \end{split}
  \end{equation}
  This shows that the claim holds with 
  $C = b_n \sqrt{\frac{\pi}{2} \, |S^{n-2}|} \, \norm[H^{1/2}]{e}$.
\end{tod}


Almost the same derivation yields an analogous result for $\Lambda e * f$:

\begin{theorem}
  \label{thm:EConvLambdaFReconstructionStability}
  If $n \ge 2$ and 
  $e \in H^{3/2}(\Rn)$ is such that $\Lambda e \in \Dxr(\Rn)$,
  then there exists a constant $C > 0$ such that 
  \begin{equation}
    |\Lambda e * f(x)| 
    \le C \, \norm[L^2(\Sm(0, R) \times \Sm)]{\DD f}
  \end{equation}
  for all $f \in L^2_0(\Rn)$ with $\supp f \subset B(0,R)$ and all $x \in \Rn$.
\end{theorem}

\begin{tod}
  As in the proof of Theorem~\ref{thm:ApproxReconstructionStability},
  \begin{equation}
%  \begin{split}
    |e * f(x)|^2
%    &
%    \le \frac{b_n^2}{4} \int_\Sm 
%      \norm[L^2(\thb)]{\laplace \PP_\theta e} \norm[L^2(\thb)]{\PP_\theta f} 
%      \, d\theta
%    \\&
    \le \frac{b_n^2}{4} 
      \int_\Sm \int_\thb |\laplace \PP_\theta e(y)|^2 \,dy \,d\theta
      \int_\Sm \int_\thb |\PP_\theta f(y)|^2 \,dy \,d\theta.
%  \end{split}
  \end{equation}
  by Theorem~\ref{thm:EConvLambdaFReconstruction},
  Young's inequality on $\thb$ and Hölder's inequality on $\Sm$.
  The first part of Theorem~\ref{thm:EConvLambdaFReconstruction} 
  and Lemma~\ref{lemma:SKlemma46} show that
  \begin{align}
    \int_\Sm \int_\thb |\laplace \PP_\theta e(y)|^2 \,dy \,d\theta
    &
    = 2\pi \int_\Sm \int_\thb \big| |\xi|^4 \, |\Fe(y)|^2 \,dy \,d\theta
    \\&
    \notag
    = 2\pi \, |S^{n-2}| \int_\Rn |\xi|^3 |\Fe(y)|^2 \,dy
    = 2\pi \, |S^{n-2}| \norm[H^{3/2}]{e},
  \end{align}
  so that the same change of variable as in the proof of 
  Theorem~\ref{thm:ApproxReconstructionStability} yields
  \begin{equation}
    |e * f(x)| 
    \le b_n \sqrt{\frac{\pi \, S^{n-2}}{2}} \, \norm[H^{3/2}]{e}
    \norm[L^2(\Sm(0, R) \times \Sm)]{\DD f},
  \end{equation}
  as claimed.
\end{tod}





\addtocontents{toc}{\protect\pagebreak}
\chapter{Equivalence of Wave Fronts}
\label{chap:EquivalenceOfWaveFronts}

\section{Wave Front of a Distribution}
\label{sec:WaveFront}


If $\Lambda f$ is to be reconstructed instead of the real x-ray
attenuation density $f$, it is important to know that these two functions
yield similar information. 
Especially the discontinuities in the density structure, indicating 
boundaries between different materials or different types of tissues,
are often of interest, and it fortunately turns out that 
$\Lambda f$ and $f$ have exactly the same discontinuities. 
More precisely, they have the same \emph{wave fronts}, which are defined
as follows: \cite{hormander}

\begin{definition}
  \label{def:WaveFront}
  Let $X \subset \Rn$ be an open set and let $f \in \Ss'(X)$ be a
  tempered distribution.
  The point $(x_0,\, \xi_0) \in X \times (\Rn \setminus \{0\})$
  does not belong to the \emph{wave front} of $f$, if there exist
  an $\eps > 0$ and 
  a function $\phi \in C^\infty_0(\Rn)$ with $\phi(x_0) \neq 0$,
  such that for each $N \in \Z_+$ there is a constant $c_N$ for which
  \begin{equation}
    \label{WaveFrontInequality}
    \left| \left(\F(\phi f)\right)(\xi) \right| 
    < c_N \left(1 + |\xi|\right)^{-N}
  \end{equation}
  whenever $\xi$ is in the cone \label{def:cone}
  \begin{equation}
    \label{WaveFrontCone}
    K_\eps(\xi_0) := \Big\{ \xi \in \Rn \setminus \{0\} \,\Big|\,
    \big| \tfrac{\xi}{|\xi|} - \tfrac{\xi_0}{|\xi_0|} \big| < \eps \Big\}.
  \end{equation}

  All other points $(x_0,\, \xi_0) \in X \times (\Rn \setminus \{0\})$
  are said to belong to the \emph{wave front} of $f$, 
  which is denoted by $\WF{f}$.

  The projection $\{ x_0 \in X \,|\, (x_0,\,\xi_0) \in \WF{f} \}$
  is called the \emph{singular support} of $f$ and denoted by
  $\singsupp f$. \label{def:singsupp}
\end{definition}

The wave front set describes the singularities of $f$, because
if $f$ is smooth in some neighbourhood $U$ of a point 
$x_0 \in X$, then for any $\phi \in C^\infty_0$ supported in $U$,
$\phi f$ is in $C^\infty_0 \subset \Ss$ and therefore also 
$\F(\phi f) \in \Ss$. 
Consequently, $(1 + |\xi|)^N \, \F(\phi f)(\xi)$ is bounded by 
some constant $c_N$ depending on the seminorms 
$\norm[\alpha, 0]{\F(\phi f)}$, $|\alpha| \le N$.

Also conversely, if \equ{WaveFrontInequality} holds, 
then $\phi f$ is smooth. 
Thus, the singular support of $f$ consists of the points $x_0 \in X$
at which $f$ is singular. For a fixed $x_0 \in \singsupp f$,
the set of points 
$\{ \xi_0 \in \R^n \setminus \{0\} \,|\, (x_0,\, \xi_0) \in \WF{f} \}$
tells in which directions $f$ is not smooth at $x_0$.
For details and proofs, see \cite{hormander}.

We shall now show that under certain assumptions on $f$, 
the distributions $\Lambda f$ and $f$ have the same wave fronts. 
This result holds under more general conditions, too, 
but to avoid using results from the theory of pseudodifferential operators, 
we shall confine our study to square-integrable functions for which
$\Lambda f$ is in the x-ray domain $\Dxr$
\cite{ramm96:radon_and_lt}.
Such functions are, of course, 
sufficient in practical tomography applications.



\section{Auxiliary Results}

A few lemmata will be needed for the proof.

First of all, the cutoff function $\phi$ above can be chosen
to be identically 1 in a neighbourhood of $x_0$:

\begin{lemma}
  \label{lemma:PhiCanBeChosenToBeOneNearX0}
  If $n \ge 2$, $X \subset \Rn$, $f \in \Ss'(X)$ and 
  $(x_0,\,\xi_0) \not\in \WF{f}$,
  then there exist an open set $U \subset X$,
  a function $\phi \in C_0^\infty$ and constants
  $\eps > 0$, $c_1,\, c_2,\, c_3,\, \ldots$ such that
  $\phi|_{U}=1$, $x_0 \in U$ and
  \begin{equation}
    \label{LemmaPhiCanBeChosenToBeOneNearX0Inequality}
    \left|\F(\phi f)(\xi)\right| < c_N \left(1 + |\xi|\right)^{-N}
    \text{ whenever } \xi \in K_\eps(\xi_0)
  \end{equation}
  for all $N \in \Z_+$.
\end{lemma}
\begin{tod}
  Since $(x_0,\,\xi_0) \in \WF{f}$, there is a function 
  $\psim \in C_0^\infty$ with $\psim(x_0) \neq 0$ and constants $\eps > 0$
  and $\tilde{c}_N$, $N \in \Z_+$, such that
  \begin{equation}
    \label{LemmaPhiCanBeChosenWaveFrontInequalityForPsim}
    \big|\F(\psim f)(\xi)\big| 
    < \tilde{c}_N \left(1 + |\xi|\right)^{-N}
  \end{equation}
  whenever $\xi \in K_\eps(\xi_0)$.
% Set $\psi(x) = \big[|\psim(x_0)| \, \psim(x_0)^{-1}\big]\, \psim(x)$
  Set
  \begin{equation}
    \label{LemmaPhiCanBeChosenDefPsi}
    \psi(x) = \frac{|\psim(x_0)|}{\psim(x_0)}\, \psim(x) 
  \end{equation}
  so that $a := \psi(x_0) = |\psim(x_0)| > 0$. Since $\psi$ is continuous, 
  an open neighbourhood $V$ of $x_0$ can be chosen such that 
  $\Re\, \psi(x) > \frac{a}{2}$ whenever $x \in V$. 
  Choose $\eps_1,\, \eps_2 > 0$ such that 
  $B(x_0,\eps_1) \subset B(x_0,\eps_2) \subset V$ and a function
  $\eta \in C^\infty_0$ with properties 
  $0 \le \eta(x) \le 1$ for all $x \in X$, $\eta|_{B(x_0,\eps_1)} = 1$
  and $\eta(x) = 0$ when $x \not\in B(x_0,\eps_2)$;
  for instance the function $\tilde{\chi}_{\eps_1, \eps_2}$
  in \equ{ApproxCharacteristicFunction} will do.
  
  Now set $g(x) = \eta(x) \psi(x) + (1 - \eta(x)) a/4$.
  This function is clearly smooth and it does not vanish anywhere:
  When $|x - x_0| < \eps_1$, $|g(x)| \ge \Re\, \psi(x) > \tfrac{a}{2}$;
  when $|x - x_0| > \eps_2$, $|g(x)| = \tfrac{a}{4} > 0$,
  and when $\eps_1 \le |x - x_0| \le \eps_2$,
  \begin{equation}
    |g(x)| \ge \Re\, g(x) 
    = \eta(x) \, \Re\, \psi(x) + (1 - \eta(x))\, \frac{a}{4}
%    = \frac{a}{4} + \eta(x) \Big(\Re\, \psi(x) - \frac{a}{4}\Big)
    \ge \frac{a}{4} > 0.
  \end{equation}
  We can therefore define the function 
  $\phi(x) = \psi(x) / g(x) \in C_0^\infty(\Rn)$
  with the property $\phi(x) = 1$ whenever $x \in U := B(x_0, \eps_1)$.

  For showing that $\phi$ satisfies 
  \equ{LemmaPhiCanBeChosenToBeOneNearX0Inequality},
  first fix $h \in C_0^\infty$ such that $h|_{\supp \psi} = 1$;
  once again, $\tilde{\chi}_{\eps_1', \eps_2'}$ can be used 
  if $\eps_1'$ is chosen so large that $\supp \psi \subset B(0,\eps_1')$ 
  and $\eps_2' > \eps_1'$.
  Then $h \psi = \psi$ and
  \begin{equation}
    \F(\phi f) = \F\left(\frac{\psi f}{g}\right)
    = \F\left(\frac{h \psi f}{g}\right)
    = \F\left(\frac{h}{g}\right) * \F(\psi f).
  \end{equation}
  Therefore,
  \begin{equation}
    \left|\F(\phi f)(\xi)\right|
%    = \left| \int_\Rn \F\left(\frac{h}{g}\right)(\zeta) \,
%               \F(\psi f)(\xi - \zeta) \,d\zeta \right|
    \le \int_\Rn \big| \F\big(\tfrac{h}{g}\big)(\zeta)\big|
                 \left| \F(\psi f)(\xi - \zeta) \right| \,d\zeta.
  \end{equation}
  Using \equ{LemmaPhiCanBeChosenWaveFrontInequalityForPsim}
  and the fact that the constant $|\psim(x_0)|/\psim(x_0)$
  in \equ{LemmaPhiCanBeChosenDefPsi} has modulus 1, yields
  \begin{equation}
    \left|\F(\phi f)(\xi)\right|
    \le \tilde{c}_N \int_\Rn 
        \big| \F\big(\tfrac{h}{g}\big)(\zeta) \big|
        \left(1 + |\xi - \zeta|\right)^{-N} \,d\zeta.
  \end{equation}
  We then split the domain of integration into two parts,
  $|\zeta| \le \tfrac{1}{2} |\xi|$ and $|\zeta| > \tfrac{1}{2} |\xi|$,
  and call the integrals over these two parts $I_1$ and $I_2$, respectively.
  We estimate them separately.

  When $|\zeta| \le \frac{1}{2}|\xi|$, 
  $|\xi - \zeta| \ge \left| |\xi| - |\zeta| \right| \ge \frac{1}{2}|\xi|$,
  and therefore 
  \begin{equation}
    \left(1 + |\xi - \zeta|\right)^{-N} 
    \le \Big(1 + \frac{1}{2}\,|\xi| \Big)^{-N}
    \le \bigg(\frac{1}{2} + \frac{1}{2}\,|\xi|\bigg)^{-N}
    = 2^N \left(1 + |\xi|\right)^{-N},
  \end{equation} which shows that
  \begin{equation}
  \label{LemmaIOneEstimate}
  \begin{split}
    I_1 
  & \le \tilde{c}_N \int_{|\zeta| \le \frac{1}{2}|\xi|}
    \big| \F\big(\tfrac{h}{g}\big)(\zeta)\big| \,
    2^N \left(1 + |\xi|\right)^{-N} \, d\zeta
  \\
  & \le 2^N \tilde{c}_N \,\bnl{\F\big(\tfrac{h}{g}\big)}{1} \,(1 + |\xi|)^{-N}.
  \end{split}
  \end{equation}
  For estimating $I_2$, note that $(1 + |\xi - \zeta|)^{-N} < 1$, whence
  \begin{equation}
    I_2
    \le \int_{|\zeta| > \frac{1}{2}|\xi|} 
            (1 + |\zeta|)^N \, (1 + |\zeta|)^{-N} \,
            \big|\F\big(\tfrac{h}{g}\big)(\zeta)\big| \,d\zeta.
  \end{equation}
  Since in the domain of integration, 
  \begin{equation}
    \left(1 + |\zeta|\right)^{-N} 
    < \Big(1 + \frac{1}{2} \, |\xi| \Big)^{-N} 
    < \left(\frac{1}{2} + \frac{1}{2} \, |\xi| \right)^{-N} 
    = 2^N\, (1 + |\xi|)^{-N},
  \end{equation}
  we see that
  \begin{equation}
  \label{LemmaITwoEstimate}
  \begin{split}
    I_2
  & \le 2^N \tilde{c}_N \int_{|\zeta| > \frac{1}{2}|\xi|} (1 + |\zeta|)^N \,
            \big|\F\big(\tfrac{h}{g}\big)(\zeta)\big| \,d\zeta \,
            (1 + |\xi|)^{-N}
  \\
  & \le 2^N \tilde{c}_N\, 
        \bnl{(1 + |\zeta|)^N \, \F\big(\tfrac{h}{g}\big)(\zeta)}{1}\,
        (1 + |\xi|)^{-N}.
  \end{split}
  \end{equation}
  
  Combining estimates \equ{LemmaIOneEstimate} and \equ{LemmaITwoEstimate}
  we see that \equ{LemmaPhiCanBeChosenToBeOneNearX0Inequality} holds,
  if we choose
  \begin{equation}
    c_N 
    = 2^N \tilde{c}_N 
            \left(\bnl{\F\big(\tfrac{h}{g}\big)}{1} 
          + \bnl{(1 + |\zeta|)^N\, \F\big(\tfrac{h}{g}\big)(\zeta)}{1}
          \right),
  \end{equation}
  which is certainly finite, as $\F(h/g) \in \Ss(\Rn)$.
\end{tod}


The geometry of two coaxial cones is described by

\begin{lemma}
  \label{lemma:WFGamma1And2Geometry}
  If $n \ge 2$, $\xi_0 \in \Rn \setminus \{0\}$ and $\eps \in (0, 1/\sqrt{2})$,
  then there is a constant $C > 0$ such that
  \begin{equation}
    |\xi - \zeta| \ge C \,|\xi|
    \qquad \text{and} \qquad
    |\xi - \zeta| \ge C \,|\zeta|
  \end{equation}
  whenever $\xi \in K_{\eps/2}(\xi_0)$ and 
  $\zeta \in \Rn \setminus K_{\eps}(\xi_0)$.
\end{lemma}

\begin{tod}
  Because $K_{\delta}(\xi_0) = K_{\delta}(\lambda \xi_0)$ for all 
  $\lambda > 0$, we can assume that $|\xi_0| = 1$.
  Write $\Km := K_{\eps}(\xi_0)$ and $\Km' := K_{\eps/2}(\xi_0)$.
  Consider the plane $T \subset \Rn$ spanned by the origin, $\xi$ and $\zeta$
  and call $\xi_0'$ the orthogonal projection of $\xi_0$ onto $T$.
  The intersection of a cone with a plane through its vertex is a sector
  of a circle, so the intersection of $T$ with the hull of $\Km$ consists of 
  two rays from the origin. Let $\kappa$ be the point located at
  distance $1$ from the origin on the closer one to $\zeta$ of these two rays,
  and let $\kappa'$ be the point with the same properties with respect
  to $\Km'$. (See Figure \ref{fig:WaveFrontGeometryTwo}.)

  \kuvienpoisto{
  \begin{figure}[htbp]
    \begin{center}
      \leavevmode
      \epsfig{file=pics/wf-geom2.eps,height=8cm}
      \caption{Geometry for constructing the lower bound $\alpha_0$ 
        in the proof of Lemma \ref{lemma:WFGamma1And2Geometry}. 
        Lines not in the plane $T$ are drawn as dashed.}
      \label{fig:WaveFrontGeometryTwo}
    \end{center}
  \end{figure}}

  Write $\mu = |\xi_0'|$,
  $\alpha = \measuredangle \kappa' 0 \kappa$,
  $\beta = \measuredangle \xi_0' 0 \kappa'$
  and $\gamma = \measuredangle \xi_0' 0 \kappa = \alpha + \beta$,
  where 
  \begin{equation}
    \measuredangle x_1 x_2 x_3 
    = \arccos \frac{(x_1 - x_2) \cdot (x_3 - x_2)}{|x_1 - x_2|\,|x_3 - x_2|}
    \in [0, \pi]
  \end{equation}
  denotes the angle between the line segments from $x_2$ to $x_1$ and $x_3$.
  \label{def:measuredangle}
  As $\zeta$ moves in $\Rn \setminus \Km$, $\mu$ varies in 
  $(\sqrt{1 - \eps^2/4}, 1]$.

  Applying the Pythagorean and cosine theorems for the triangles 
  $\xi_0\xi_0'\kappa'$ and $\xi_0' 0 \kappa'$, respectively, yields
  \begin{equation}
    \frac{\eps^2}{4} - (1^2 - \mu^2) = \mu^2 + 1^2 - 2 \mu \cos \beta,
  \end{equation}
  and for $\xi_0 \xi_0' \kappa$ and $\xi_0' 0 \kappa$, yields
  \begin{equation}
     \eps^2 - (1^2 - \mu^2) = \mu^2 + 1^2 - 2 \mu \cos \gamma,
  \end{equation}
  and consequently
  \begin{equation}
    \alpha = \gamma - \beta
    = \arccos \frac{2 - \eps^2}{2\mu} - \arccos \frac{8 - \eps^2}{8\mu}.
  \end{equation}
  A few calculations
  show that given $\eps \in (0, 1/\sqrt{2})$, 
  this is a decreasing function of $\mu \in (\sqrt{1 - \eps^2/4}, 1]$
  and has a lower bound $\alpha_0 > 0$ such that 
  \begin{equation}
    \alpha \ge \alpha_0
    \qquad \text{for all } \mu \in (\sqrt{1 - \eps^2/4}, 1].
  \end{equation}

  Denote by $\zeta'$ the point which is further away from $\xi_0$
  of the two points of $T$
  with the properties $|\xi - \zeta'| = |\xi - \zeta|$
  and $\zeta' - \xi \,\bot\, \xi$.

  \kuvienpoisto{
  \begin{figure}[htbp]
    \begin{center}
      \leavevmode
      \epsfig{file=pics/wf-geom1.eps,height=8cm}
      \caption{Geometry in the proof of the first part of 
        Lemma \ref{lemma:WFGamma1And2Geometry}.}
      \label{fig:wf-geom1}
    \end{center}
  \end{figure}}

  Figure \ref{fig:wf-geom1} depicts this situation and
  demonstrates that since $\alpha$ is clearly at most as great
  as the aperture of $K_\eps(\xi_0)$, which is less that $\pi/2$,
  \begin{equation}
    \frac{|\xi - \zeta|}{|\xi|} = \frac{|\xi - \zeta'|}{|\xi|}
    \ge \frac{|\xi - \zeta'|}{b} \ge \frac{a}{b} 
    \ge \frac{a'}{b} = \sin\alpha \ge \sin\alpha_0 =: C
  \end{equation}
  for all $\xi \in \Km'$, $\zeta \in \Rn \setminus \Km$, 
  proving the first claim.

  For the second one, write $\delta = \measuredangle \zeta 0 \xi$
  and distinguish two cases according to whether $\delta$ is 
  greater or smaller than $\pi/2$.

  If $\delta < \pi/2$, the situation is as in Figure~\ref{fig:wf-geom3}, 
  and we see that
  \begin{equation}
    \label{WFGamma2Part1}
    \frac{|\xi - \zeta|}{|\zeta|} 
    \ge \frac{\tilde{a}}{|\zeta|}
    \ge \frac{\tilde{b}}{|\zeta|}
    = \sin(\alpha + \beta_1)
    \ge \sin \alpha
    \ge \sin \alpha_0.
  \end{equation}
  \kuvienpoisto{
  \begin{figure}[htbp]
    \begin{center}
      \leavevmode
      \epsfig{file=pics/wf-geom3.eps,width=9cm}
      \caption{Geometry in the proof of the second part of 
        Lemma \ref{lemma:WFGamma1And2Geometry}.}
      \label{fig:wf-geom3}
    \end{center}
  \end{figure}}

  If then $\delta \ge \pi/2$, simply denote by $\xi'$
  the point on the line joining $\xi$ and $\zeta$ that is
  perpendicular to $\zeta$. 
  Figure~\ref{fig:wf-geom4} demonstrates that 
  \begin{equation}
    \label{WFGamma2Part2}
    |\zeta - \xi|
    = |\zeta - \xi'| + |\xi' - \xi|
    \ge |\zeta - \xi'|
    =   \sqrt{|\xi'|^2 + |\zeta|^2}
    \ge |\zeta|
    \ge \sin\alpha_0 \, |\zeta|.
  \end{equation}

  \kuvienpoisto{
  \begin{figure}[htbp]
    \begin{center}
      \leavevmode
      \vspace{1ex}
      \epsfig{file=pics/wf-geom4.eps,width=11.5cm}
      \caption{Geometry in the proof of the second part of 
        Lemma \ref{lemma:WFGamma1And2Geometry}.}
      \label{fig:wf-geom4}
    \end{center}
  \end{figure}}

  Combining estimates \equ{WFGamma2Part1} and \equ{WFGamma2Part2}
  shows that the second claim holds, too.
\end{tod}



Before proceeding to show that $\Lambda f$ and $f$ have the same
wave fronts, we shall derive the following result.
It will be used for seeing that when $f$ is decomposed
as $(1 - \phi) f + \phi f$, the former term does not produce singularities
in $\Lambda f$ or $R_1 * f$:

\begin{lemma}
  \label{lemma:Lambda1MinusPhiFIsSmooth}
  Let $n \ge 2$, let $U \subset \Rn$ be open and let
  $\eta, \phi \in C^\infty_0(\Rn)$
  be such that $\supp \eta \subset U$ and $\phi|_U = 1$.
  If $f \in \Dxr(\Rn)$ and $R_1 * f \in L^2(\Rn)$, then
  \begin{equation}
    \eta \, R_1 * \big[(1 - \phi) f\big] \in C^\infty_0(\Rn),
  \end{equation}
  and if $f \in L^2(\Rn)$ and $\Lambda f \in L^2(\Rn)$, then
  \begin{equation}
    \eta \, \Lambda \big[(1 - \phi) f\big] \in C^\infty_0(\Rn).
  \end{equation}
\end{lemma}
\begin{tod}
  Since $\eta$ has bounded support, so do the two functions in the claims.
  For proving that
  \begin{equation}
    \label{R1Conv1MinusPhiFExpression}
    \eta(x) R_1 * \big[(1 - \phi) f\big] (x)
    = b_n \eta(x) \int_\Rn \frac{1 - \phi(y)}{|x - y|^{n-1}} \, f(y) \,dy
  \end{equation}
  is smooth, note that in order 
  for both $1 - \phi(y)$ and $\eta(x)$ to be non-zero,
  \begin{equation}
    |x - y| \ge \dist \left(\supp \eta,\, \supp (1-\phi)\right) =: d_0
  \end{equation}
  which is positive because $\supp (1 - \phi) \subset \Rn \setminus U$
  and $\supp \eta \subset U$ are disjoint closed sets
  (see Figure~\ref{fig:supports}).

  \begin{figure}[htbp]
    \begin{center}
      \leavevmode
      \epsfig{file=pics/supports.eps, width=8cm}
      \caption{The distance between the supports of $\eta$ and $1 - \phi$ 
        in the proof of Lemma~\ref{lemma:Lambda1MinusPhiFIsSmooth} 
        is positive.}
      \label{fig:supports}
    \end{center}
  \end{figure}

  The integrand in \equ{R1Conv1MinusPhiFExpression} is therefore
  nonsingular, and the Leibniz rule gives
  \begin{multline}
    \partial^\alpha \big(\eta(x) R_1 * \left[(1 - \phi) f\big] (x) \right)
    \\
    = b_n \sum_{\beta \le \alpha} c_{\alpha\beta} \,
      \partial^{\alpha - \beta} \eta(x) 
      \int_{\Rn \setminus B(x_0, d_0)} \partial^\beta_x 
        \frac{1 - \phi(y)}{|x - y|^{n-1}} \,f(y) \,dy.
  \end{multline}
  Differentiation under the integral sign is permitted because,
  with $h := (1 - \phi)\, f \in L^2$,
  \begin{equation}
  \begin{split}
    \partial_k \int_{\Rn \setminus B(x_0,d_0)} &\frac{h(y)}{|x - y|^{n-1}} \,dy
    \\&
    = \lim_{\eps \to 0} \int_{\Rn \setminus B(x_0,d_0)} \frac{h(y)}{\eps}
      \left[ \frac{1}{|x + \eps e_k - y|^{n-1}} - \frac{1}{|x - y|^{n-1}} 
      \right] \,dy
    \\&
    = \int_{\Rn \setminus B(x_0,d_0)} \partial_k \, \frac{h(y)}{|x - y|^{n-1}}
    \, dy
  \end{split}
  \end{equation}
  by Lebesgue's theorem of dominated convergence:
  When $|\eps| \le d_0/2$, 
  \begin{equation}
    |x + \eps e_k - y| 
    \ge \big| |x - y| - |\eps| \big| 
    \ge d_0 - \frac{d_0}{2} 
    = \frac{d_0}{2}
    > 0,
  \end{equation}
  and therefore $|x - y|^{1-n}$ is smooth, so that the mean value theorem
  yields
  \begin{equation}
  \begin{split}
    \left[\frac{1}{|x + \eps e_k - y|^{n-1}} - \frac{1}{|x - y|^{n-1}} \right]
    &
    =  \dodo{}{\tilde{x}_k} \frac{1}{|\tilde{x} - y|^{n-1}}
      \Big|_{\tilde{x} = x + \xi e_k} \eps
    \\&
    = \frac{x_k + \xi - y_k}{|x + \xi e_k - y|^{n+1}} \, \eps
  \end{split}
  \end{equation}
  for some $\xi \in \R$ with $|\xi| < |\eps|$. The fact that 
  $|\xi| < |\eps| \le d_0/2 \le |x - y|/2$ also implies that
  \begin{equation}
    |x + \xi e_k - y| 
    \ge \big| |x - y| - |\xi| \big|
    \ge |x - y| - \frac{|x - y|}{2}
    = \frac{|x - y|}{2},
  \end{equation}
  which yields
  \begin{equation}
  \begin{split}
    \left|\frac{h(y)}{\eps} \left[\frac{1}{|x + \eps e_k - y|^{n-1}}
          - \frac{1}{|x - y|^{n-1}} \right] \right|
    &
    = \left|\frac{h(y)}{\eps} \, 
      \frac{x_k + \xi - y_k}{|x + \xi e_k - y|^{n+1}} \, \eps \right|
    \\&
    \le \left| \frac{h(y)}{|x + \xi e_k - y|^n} \right|
    \\&
    \le \frac{2^n \, |h(y)|}{|x - y|^n}
    =: g(y).
  \end{split}
  \end{equation}
  The dominant function $g$ is integrable because
  \begin{equation}
    \nl{g}{1}^2 
    \le 2^{2n} \, |\Sm| \int_{d_0}^\infty r^{-2n} \, r^{n-1} \,dr
      \, \nl{h}{2}^2
    < \infty.
  \end{equation}
  The same procedure also works inductively for higher-order derivatives, 
  since
  \begin{multline}
    \partial^{\gamma + e_k} \int_{\Rn \setminus B(x_0,d_0)}
      \frac{h(y)}{|x - y|^{n-1}} \,dy
    \\
    = \lim_{\eps \to 0} \int_{\Rn \setminus B(x_0,d_0)}
      \left[ \partial^\gamma R_1(x + \eps e_k - y) - \partial^\gamma R_1(x - y)
        \right] \,dy
  \end{multline}
  and here the integral is dominated in absolute value by
  \begin{equation}
    \left| \frac{h(y)}{\eps} \,
      \partial^{\gamma + \eps_k} R_1(x - \xi e_k - y) \, \eps \right|
    \le \frac{2^{n + |\gamma|} \, |h(y)|}{|x - y|^{n + |\gamma|}}
    \in L^1,
    \quad \quad
    |\xi| < |\eps| \le \frac{d_0}{2}.
  \end{equation}
  This shows that $\eta \, R_1 * \big[(1 - \phi) f\big] \in C^\infty(\Rn)$.

  Analogously for the second part of the claim,
  \begin{align}
    \partial^\alpha \Big(\eta(x & ) \,\Lambda \big[(1 - \phi) f\big](x)\Big)
    \notag
    \\&
    = \partial^\alpha \Big(
      - \eta(x) \sum_{j=1}^n \pv \dodo{R_1}{x_j} * D_j \big[(1 - \phi)f\big](x)
      \Big)
    \\&
    \notag
    = - \sum_{\beta \le \alpha} c_{\alpha\beta} \,
      \partial^{\alpha - \beta} \eta(x) \sum_{j=1}^n
      \int_{\Rn \setminus B(x_0,d_0)}
        \partial^{e_j + \beta}_x R_1(x - y)
        \, D_j \big[(1 - \phi) f\big](y) \,dy.
  \end{align}
  This time the induction step is
  \begin{multline}
    \partial^{\gamma + e_k} \int_{\Rn \setminus B(x_0,d_0)}
      \dodo{R_1}{x_j} (x - y) \, D_j h(y) \,dy
    \\
    = \lim_{\eps \to 0} \int_{\Rn \setminus B(x_0,d_0)}
      \frac{D_j h(y)}{\eps} \left[ \partial^{e_j + \beta} R_1(x + \eps e_k - y)
        - \partial^{e_j + \beta} R_1(x - y) \right]
        \, dy.
  \end{multline}
  Now the integrand is dominated in absolute value by
  \begin{equation}
  \begin{split}
    \left|\frac{D_j h(y)}{\eps} \, 
      \partial^{e_j + \beta + e_k} R_1(x + \xi e_k - y) \, \eps \right|
    &
    \le \frac{|D_j h(y)|}{|x + \xi e_k - y|^{n - 1 + 1 + |\beta| + 1}}
    \\&
    \le \frac{2^{n + |\beta| + 1} \, |D_j h(y)|}{|x - y|^{n + |\beta| + 1}}
    =: g(y),
  \end{split}
  \end{equation}
  which is integrable since
  \begin{equation}
    \nl{g}{1}^2
    \le 2^{2n + 2|\beta| + 2} \, |\Sm| 
      \int_{d_0}^\infty r^{-2n-2|\beta|-2} \, r^{n-1} \,dr \,
      \nl{D_j h}{2}^2
  \end{equation}
  and
  \begin{equation}
    \nl{D_j h}{2}
    = \nl{(D_j \phi) f + (1 - \phi) D_j f}{2}
    \le \nl{D_j \phi}{\infty} \nl{f}{2} + \nl{1 - \phi}{\infty} \nl{D_j f}{2}
  \end{equation}
  is finite
  because $f \in L^2$ and $\nl{D_j f}{2} = \nl{i \xi_j \Ff}{2}
  \le \nl{|\xi| \, \Ff}{2} = \nl{\Lambda f}{2} < \infty$.
\end{tod}



\section{$\Lambda f$ and $f$ Have Equal Wave Fronts}
\label{sec:WFLambdaFEqWFF}

We are now ready to present the main result of this chapter:

\begin{theorem}
  \label{thm:WaveFrontsEquivalent}
  If $n \ge 2$,
  $f \in L^2(\Rn)$ and $\Lambda f \in \Dxr(\Rn)$, 
  then $\WF{f} = \WF{\Lambda f}$.%
\end{theorem}
\begin{tod}
  First for showing the inclusion $\WF{\Lambda f} \subset \WF{f}$,
  choose $(x_0,\,\xi_0) \not\in \WF{f}$ 
  and show that $(x_0,\,\xi_0) \not\in \WF{\Lambda f}$.
  The converse inclusion will be shown later.
  Fix $U$, $\phi$, $\eps$, $c_1$, $c_2$, $c_3$, $\ldots$ 
  according to Lemma~\ref{lemma:PhiCanBeChosenToBeOneNearX0}.
  Assume that $\eps < 1/\sqrt{2}$, because if the condition
  \equ{LemmaPhiCanBeChosenToBeOneNearX0Inequality}
  holds for an $\eps \ge 1/\sqrt{2}$, it holds for a smaller one as well.

  Select a function $\eta \in C^\infty_0(\Rn)$ 
  such that $\supp \eta \subset U$.
  We shall prove that there are constants 
  $\eps' > 0$, $c_1',\, c_2',\, c_3',\, \ldots$
  such that
  \begin{equation}
    \left|\F(\eta \Lambda f)(\xi)\right| < c_N' \left(1 + |\xi|\right)^{-N}
  \end{equation}
  for all $N \in \Z_+$, whenever $\xi \in K_{\eps'}(\xi_0)$.
  To this end, write 
  $\Lambda f = \Lambda(\phi f) + \Lambda\left((1-\phi) f\right)$.
  By Lemma~\ref{lemma:Lambda1MinusPhiFIsSmooth},
  $\eta \Lambda\left((1-\phi) f\right) \in C^\infty_0$,
  which implies that 
  $\F\left(\eta \Lambda \left((1-\phi) f\right)\right) \in \Ss$.
  Therefore, $(1 + |\xi|)^N \F(\eta \Lambda((1 - \phi) f))$ is bounded
  for all $N \in \Z_+$.

  It is therefore sufficient to find constants 
  $\eps' > 0,\, c_1',\, c_2',\, c_3' \ldots\,$ such that
  \begin{equation}
    |\F(\eta \Lambda(\phi f))(\xi)| < c_N' (1 + |\xi|)^{-N}
  \end{equation}
  for all $N \in \Z_+$ and $\xi \in K_{\eps'}(\xi_0)$. In fact, because 
  $1 + |\xi|^N > \frac{1}{2}\,(1 + |\xi|)^N$ for large $|\xi|$,
  say for $|\xi| > R_0$, and because
  \begin{equation}
    d_N = \max_{|\xi| \le R_0} \frac{(1 + |\xi|)^N}{1 + |\xi|^N} < \infty,
  \end{equation}
  we see that always
  \begin{equation}
    \big( 1 + |\xi|^N \big)^{-1} \le \max \{d_N,\, 2\} \, (1 + |\xi|)^{-N}.
  \end{equation}
  Hence it suffices to find constants $c_N'$ such that
  \begin{equation}
    \label{WaveFrontsEqualSufficesToShow}
    |\F(\eta \Lambda(\phi f))(\xi)| < c_N' \left(1 + |\xi|^N\right)^{-1}
  \end{equation}
  whenever $\xi \in K_{\eps'}(\xi_0)$ for some $\eps' > 0$,
  or equivalently to show that for each $N$,
  \begin{equation}
    \left(1 + |\xi|^N\right) |\F(\eta \Lambda(\phi f))(\xi)|
  \end{equation}
  is bounded as $\xi$ moves in $K_{\eps'}(\xi_0)$.
  This is what will be done next, with $\eps' = \eps/2$.

  Choose $N \in \Z_+$ arbitrarily and examine the expression 
  \begin{equation}
    \F(\eta \Lambda(\phi f))(\xi) = \F\eta * \F(\Lambda(\phi f))(\xi)
    = \int_\Rn \Feta(\xi - \zeta)\, |\zeta|\, \F(\phi f)(\zeta) \,d\zeta.
  \end{equation}
  Split the domain of integration into two parts, the cone 
  $\Km := K_{\eps}(\xi_0)$, and its complement.
  Using the triangle inequality, we obtain
  \begin{equation}
    \left(1 + |\xi|^N\right) |\F(\eta \Lambda(\phi f))(\xi)| 
    \le I_1(\xi) + I_2(\xi),
  \end{equation}
  where
  \begin{align}
    I_1(\xi) &= \left(1 + |\xi|^N\right) \int_{\Km} 
             |\Feta(\xi - \zeta)|\,|\zeta|\, |\F(\phi f)(\zeta)|\, d\zeta
    \\
  \intertext{and}
    I_2(\xi) &= \left(1 + |\xi|^N\right) \int_{\Rn \setminus \Km}
             |\Feta(\xi - \zeta)|\,|\zeta|\,|\F(\phi f)(\zeta)|\, d\zeta.
  \end{align}
  We shall show that both $I_1(\xi)$ and $I_2(\xi)$ are bounded.

  For estimating $I_1$, note that since $\Feta \in \Ss(\Rn)$,
  \begin{equation}
    \label{WaveFrontTheoremGamma3}
    \Feta(\xi - \zeta) \le C_1 (1 + |\xi - \zeta|^N)^{-1}
  \end{equation}
  for some $C_1 > 0$, and that by the choice of $\phi$,
  \begin{equation}
    |\F(\phi f)(\zeta)| 
    \le c_{N+n+2} (1 + |\zeta|)^{-N-n-2} 
    \le c_{N+n+2} (1 + |\zeta|^{N+n+2})^{-1}.
  \end{equation}
  Therefore
  \begin{equation}
    I_1(\xi)
%    \le \int_{\Km} C_1 & \left(1 + |\xi-\zeta|^N\right)^{-1} \, 
%        |\zeta| \, c_{N+n+2} \left(1 + |\zeta|^{N+n+2}\right)^{-1} \,d\zeta.
%        \left(1 + |\xi|^N\right)
    \le C_1 \, c_{N+n+2}
        \int_{\Km} \frac{1 + |\xi|^N}{1 + |\xi - \zeta|^N}
          \frac{|\zeta|}{1 + |\zeta|^{N+n+2}} \,d\zeta.
  \end{equation}
  To see that this is bounded, observe that
  \begin{equation}
  \begin{split}
    \label{WaveFrontsEqualBinomialExpansion}
    \frac{1 + |\xi|^N}{1 + |\xi - \zeta|^N}
  & = \frac{1 + |(\xi - \zeta) + \zeta|^N}{1 + |\xi - \zeta|^N}
    \le \frac{1 + (|\xi - \zeta| + |\zeta|)^N}{1 + |\xi - \zeta|^N}
  \\&
    = \frac{1 + |\xi - \zeta|^N}{1 + |\xi - \zeta|^N}
      + \sum_{j = 1}^N \binom{n}{j} 
          \frac{|\xi - \zeta|^{N-j} |\zeta|^j}{1 + |\xi - \zeta|^N}
  \\&
    = 1 + \sum_{j=1}^N \binom{n}{j} \,|\zeta|^j \,
            \frac{1}{|\xi - \zeta|^{j - N} + |\xi - \zeta|^j}
  \\&
    \le 1 + \sum_{j = 1}^N \binom{n}{j}\, |\zeta|^j
    = (1 + |\zeta|)^N,
  \end{split}
  \end{equation}
  which shows that
  \begin{equation}
  \begin{split}
    I_1(\xi)
  &
%    = C_1 \, c_{N+n+2}
%      \int_{\Km} \frac{(1 + |\zeta|)^N\, |\zeta|}{1 + |\zeta|^{N+n+2}} 
%          \,d\zeta
    \le C_1 \, c_{N+n+2} \int_{\Rn} 
        \frac{(1 + |\zeta|)^N\, |\zeta|}{1 + |\zeta|^{N+n+2}} \,d\zeta
  \\&
    = C_1 \, c_{N+n+2} \, |\Sm|
        \int_0^\infty \frac{(1 + r)^N\,r\,r^{n-1}}{1 + r^{N+n+2}}\,dr.
  \end{split}
  \end{equation}
  The integrand is majorized on $[0,1]$ by
  $2^N 1^n / (1 + 0^{N+n+2}) = 2^N$ and on $[1,\infty)$ by
  $(2r)^N r^n / r^{N+n+2} = 2^N r^{-2}$, both of which are integrable
  over the respective intervals.
  Therefore $I_1(\xi)$ is bounded.

  For $I_2$, use the fact that as $\Feta \in \Ss$,
  \begin{equation}
    \label{WaveFrontTheoremGamma4}
    \Feta(\xi - \zeta) \le C_2 (1 + |\xi - \zeta|)^{-N-n-2}
  \end{equation}
  for some $C_2 > 0$.
%  , which together with
%  the first estimate of Lemma~\ref{lemma:WFGamma1And2Geometry}
%  shows that
%  whenever $\xi \in \Km'$ and $\zeta \in \Rn \setminus \Km$,
%  \begin{equation}
%    \Feta(\xi - \zeta) \le C_5 (1 + |\xi|)^{-N-n-2},
%  \end{equation}
%  where $C_5 = C_2 \, \max\{ 1, C^{-N-n-2} \}$.
  Also,
  \begin{equation}
  \begin{split}
    \label{WaveFrontsEqualFPhiFBounded}
    |\F(\phi f)(\xi)| 
    &
    \le \Fm \int_\Rn |e^{-ix \cdot \xi}|\, |\phi(x)| \, |f(x))| \, dx
    \\&
    = \Fm \nl{\phi f}{1} \le \Fm \nl{f}{2} \nl{\phi}{2} =: C_3,
  \end{split}
  \end{equation}
  so that
  \begin{equation}
  \label{WaveFrontTheoremI2Estimate}
  \begin{split}
    I_2(\xi) 
    \le \left(1 + |\xi|^N\right)
        \int_{\Rn \setminus \Km} 
        C_2 \left(1 + |\xi - \zeta|\right)^{-N-n-2} \,
        |\zeta| \,C_3 \,d\zeta.
  \end{split}
  \end{equation}
  Analyse each factor of 
  \begin{equation}
    (1 + |\xi - \zeta|)^{-N-n-2} 
    = (1 + |\xi - \zeta|)^{-1} (1 + |\xi - \zeta|)^{-N} 
      (1 + |\xi - \zeta|)^{-n-1}
  \end{equation}
  separately. First of all, the second claim of 
  Lemma~\ref{lemma:WFGamma1And2Geometry} asserts that
  \begin{equation}
    (1 + |\xi - \zeta|)^{-1} 
    \le |\xi - \zeta|^{-1} 
    \le C^{-1} \, |\zeta|^{-1},
  \end{equation}
  which cancels out the factor $|\zeta|$ 
  in \equ{WaveFrontTheoremI2Estimate}. 
  By the first claim of Lemma~\ref{lemma:WFGamma1And2Geometry}, 
  the second factor
  \begin{equation}
    (1 + |\xi - \zeta|)^{-N} 
    \le \max \{1, C^{-N} \} \, (1 + |\xi|)^{-N}
    \le \max \{1, C^{-N} \} \, (1 + |\xi|^N)^{-1}
  \end{equation}
  cancels out the factor $1 + |\xi|^N$ showing the required speed of decrease.
  The third factor is dominated by something that leaves the integral finite:
  \begin{equation}
  \begin{split}
    (1 + |\xi - \zeta|)^{-n-1} 
    &
    \le \max\{ 1, C^{-n-1} \} \, (1 + |\zeta|)^{-n-1}
    \\&
    \le \max\{ 1, C^{-n-1} \} \, (1 + |\zeta|^{n+1})^{-1}.
  \end{split}
  \end{equation}
  Therefore,
  \begin{equation}
  \begin{split}
    I_2(\xi) 
    &
    \le C_4 \left(1 + |\xi|^N\right)
    \int_{\Rn \setminus \Km}
      \frac{|\zeta|}{|\zeta| \left(1 + |\xi|^N\right)
                     \left(1 + |\zeta|^{n+1}\right)} \, d\zeta
    \\&
    \le C_4 \, |\Sm|\,\int_0^\infty \frac{r^{n-1} \,dr}{1 + r^{n+1}}
    \\&
    \le C_4 \, |\Sm|\, 
      \left(\int_0^1 \frac{dr}{1 + 0^{n+1}}
      + \int_1^\infty \frac{dr}{r^2} \right) 
%    \\&
%    = 2 C_4 \, |\Sm| 
    < \infty.
  \end{split}
  \end{equation}
  This completes the proof that $\WF{\Lambda f} \subset \WF{f}$.



  For the converse inclusion, choose $(x_0, \xi_0) \not\in \WF{\Lambda f}$
  and show that $(x_0, \xi_0) \not\in \WF{f}$. 
  Write $\psi = \Lambda f \in \Dxr$ so that
  \begin{equation}
    |\xi|^{-1} \widehat{\psi}(\xi) 
    = |\xi|^{-1} \, |\xi| \, \Ff(\xi)
    = \Ff(\xi)
    \in L^2 \subset \Lloc,
  \end{equation}
  and consequently
  \begin{equation}
%    f 
%    = \Fi(\Fp\, |\xi|^{-1} \, |\xi| \, \Fm \, |\xi| \, f) 
%    = R_1 * \Lambda f = R_1 * \psi.
%
%    \F(R_1 * \psi)(\xi) 
%    = |\xi|^{-1} \widehat{\psi}(\xi) 
%    = |\xi|^{-1} \,|\xi|\,\Ff(\xi) 
%    = \Ff(\xi)
    f 
    = \Fi\big(|\xi|^{-1} \, |\xi| \Ff(\xi)\big)
    = \Fi\big(|\xi|^{-1} \, \widehat{\psi}(\xi)\big)
    = R_1 * \psi
  \end{equation}
  by Theorem~\ref{thm:FourierTransformOfRiestPotentialInDxr}.
  Choose $U \subset \Rn$, $\phi \in C^\infty_0$,
  $\eps \in (0, 1/\sqrt{2})$ and constants $c_1$, $c_2$, \ldots,
  according to Lemma~\ref{lemma:PhiCanBeChosenToBeOneNearX0}, as before,
  but now with $f$ replaced by $\psi$.
  Then choose any $\eta \in C^\infty_0$ with $\supp \eta \subset U$
  and proceed as follows to find constants $c_1'$, $c_2'$, \ldots, such that
  \begin{equation}
    \left| \F(\eta R_1 * \psi)(\xi)\right| < c_N' (1 + |\xi|)^{-N},
  \end{equation}
  or equivalently
  \begin{equation}
    \label{WaveFrontsEqualEstimateForEtaF}
    \left| \F(\eta R_1 * \psi)(\xi)\right| < c_N' (1 + |\xi|^N)^{-1},
  \end{equation}
  for all $N \in \Z_+$ and $\xi \in K_{\eps/2}(\xi_0)$.
  As before, it suffices to show this for 
  $\eta R_1 * (\phi \psi)$ instead of $\eta R_1 * \psi$,
  because 
  \begin{equation}
    \label{WaveFrontsEqualConverseInclusionDecomposition}
    \eta R_1 * \psi
    = \eta R_1 * \big[(1 - \phi) \, \psi \big] + \eta R_1 * (\phi \psi)
  \end{equation}
  and $\eta R_1 * [(1 - \phi) \, \psi] \in C^\infty_0(\Rn)$
  by Lemma~\ref{lemma:Lambda1MinusPhiFIsSmooth},
  so that $\F(\eta R_1 * [(1 - \phi) \, \psi]) \in \Ss(\Rn)$
  and the estimate \equ{WaveFrontsEqualEstimateForEtaF} holds with
  $\eta R_1 * \psi$ replaced by $\eta R_1 * [(1 - \phi) \psi]$.

  Once again, use the fact that
  \begin{equation}
  \begin{split}
  \label{WaveFrontsEqualConverseInclusionTriangleInequality}
    \left| \F \left(\eta R_1 * (\phi \psi) \right)(\xi) \right|
    &
    = \left| \Feta * \left(|\xi|^{-1} \, \F(\phi \psi) \right) (\xi) \right|
    \\&
    \le \int_\Rn \left|\Feta(\xi - \zeta)\right|
          |\zeta|^{-1} \left| \F(\phi \psi)(\zeta) \right| \, d\zeta
  \end{split}
  \end{equation}
  and split the range of integration into two parts,
  $\Km = K_\eps(\xi_0)$ and $\Rn \setminus \Km$.
  For estimating the integral over $\Km$,
  note that since $\Feta \in \Ss$, 
  $|\Feta(\xi - \zeta)| \le C_5 (1 + |\xi - \zeta|^N)^{-1}$
  for some $C_5 > 0$, so that \equ{WaveFrontsEqualBinomialExpansion} yields
  \begin{equation}
  \label{WaveFrontsEqualConverseInclusionIntKmEstimate}
  \begin{split}
    \big(1 + |\xi|^N \big)
    &
    \int_{\Km} 
      \left|\Feta(\xi - \zeta)\right|
      |\zeta|^{-1} \left| \F(\phi \psi)(\zeta) \right| \, d\zeta
    \\&
    \le \left(1 + |\xi|^N \right) \int_{\Km} 
      C_5 \left(1 + |\xi - \zeta|^N\right)^{-1} \, |\zeta|^{-1}
      c_{N+n} \left( 1 + |\zeta|\right)^{-N-n} \, d\zeta
    \\&
    \le C_5 \, c_{N+n} \int_{\Km} 
        \frac{(1 + |\zeta|)^N}{|\zeta| \, (1 + |\zeta|)^{N+n}} \,d\zeta
    \\& 
    \le C_5 \, c_{N+n} 
        \int_\Rn \frac{1}{|\zeta| \, (1 + |\zeta|)^n} \,d\zeta
    \\& 
    = C_5\, c_{N+n} \, |\Sm| \int_0^\infty \frac{r^{n-2}}{(1 + r)^n} \,dr
    \\& 
    \le C_5 \, c_{N+n} \, |\Sm| 
        \left(\int_0^1 dr + \int_1^\infty \frac{r^{n-2}}{r^n} \, dr \right)
%    \\& 
%    = 2 \, C_5 \, c_{N+n} \, |\Sm|
    < \infty.
  \end{split}
  \end{equation}
  In the integral over $\Rn \setminus \Km$, use the fact that $\Feta \in \Ss$
  together with Lemma~\ref{lemma:WFGamma1And2Geometry} to see that
  \begin{equation}
  \begin{split}
    |\Feta(\xi - \zeta)| 
    &
    \le C_6 (1 + |\xi - \zeta|)^{-N} (1 + |\xi - \zeta|)^{-n}
    \\&
    \le C_{7} (1 + |\xi|)^{-N} (1 + |\zeta|)^{-n}
    \\& 
    \le C_{7} \left(1 + |\xi|^N \right)^{-1} (1 + |\zeta|)^{-n}.
  \end{split}
  \end{equation}
  whenever $\xi \in K_{\eps/2}(\xi_0)$.
  This estimate and the analogue of \equ{WaveFrontsEqualFPhiFBounded} yield
  \begin{equation}
  \begin{split}
  \label{WaveFrontsEqualConverseInclusionIntRnMinusKmEstimate}
    \left(1 + |\xi|^N\right)
    \int_{\Rn \setminus \Km} 
    & 
      |\Feta(\xi - \zeta)| 
      |\zeta|^{-1} \left| \F(\phi \psi)(\zeta) \right| \,d\zeta
    \\&
    \le C_{7} \left(1 + |\xi|^N\right) \int_{\Rn}
      \frac{\Fm \nl{\phi}{2} \nl{\psi}{2}}{\left(1 + |\xi|^N\right) 
                                 (1 + |\zeta|)^n \, |\zeta|} \,d\zeta
    \\&
    \le C_{7} \, \Fm \, \nl{\phi}{2} \, \nl{\psi}{2} \,|\Sm| 
      \int_0^\infty \frac{r^{n-2}}{(1 + r)^n} \,dr
    < \infty.
  \end{split}
  \end{equation}
  Combining estimates \equ{WaveFrontsEqualConverseInclusionIntKmEstimate} and 
  \equ{WaveFrontsEqualConverseInclusionIntRnMinusKmEstimate} with 
  \equ{WaveFrontsEqualConverseInclusionTriangleInequality}
  shows that $(x_0, \xi_0) \not\in \WF{f}$.
\end{tod}




\chapter{Further Properties of $\Lambda f$ and Conclusions}
\label{chap:FurtherPropertiesForLambdaF}
\label{chap:conclusions}

Even though $\WF{f} = \WF{\Lambda f}$,
jumps of $\Lambda f$ might of course be much smaller than those of $f$. 
In this case, discontinuities of $f$ would perhaps not be visible 
in $e * \Lambda f$.
Fortunately, discontinuities are clearly visible, as the following results
show.

Because at least for $f \in \Ss$
\begin{equation}\nt
  \F (\Lambda^2 f) 
  = |\xi|^2\, \Ff
  = - \sum_{j=1}^n (i\,\xi_j)^2 \Ff
  = \F (- \laplace f),
\ent\end{equation}
%\begin{equation}\nt
%  \ip{\Lambda f}{\Lambda g} = \ip{\widehat{\Lambda f}}{\widehat{\Lambda g}}
%  = \ip{|\xi|\, \Ff}{|\xi|\, \widehat{g}}
%  = \ip{\Ff}{|\xi|^2 \widehat{g}} 
%  = \ip{\Ff}{\widehat{-\laplace g}}
%  = \ip{f}{-\laplace g},
%\ent\end{equation}
the operator $\Lambda$ is sometimes called the 
square root of the negative Laplacian, \linebreak
``$\Lambda = \sqrt{\smash[b]{-\laplace}}$''.
By emphasizing edges, it indeed behaves like a first order differential 
operator. Tomography objects are often made up of areas of
constant density, which makes detecting their edges an important objective
in many applications.

Because a typical x-ray attenuation coefficient function $f$ 
can be thought of as a linear combination of characteristic functions, 
\begin{equation}
  f = \sum_{j=1}^n c_j \, \chi_{X_j},
\end{equation}
and because only linear operations on $f$ are considered,
studying $\Lambda \chi_X$ gives some insight into the behaviour of $\Lambda$.
It turns out that if $X$ is sufficiently regular, then
$\Lambda \chi_X$ behaves like $\pm 1/\dist(x, \partial X)$
near the edges of $X$, being positive inside $X$ and negative outside,
and is cupped inside $X$; see Figure~\ref{fig:lambdachi}(a).
\cite{faridani92:local}

\kuvienpoisto{
\begin{figure}[htbp]
  \begin{center}
    \subfigure[$\Lambda \chi_X$]{\epsfig{file=pics/lambdachi.eps, width=6cm}}
    \subfigure[$R_1 * \chi_X$]{\epsfig{file=pics/r1convchi.eps, width=6cm}}
    \subfigure[$L \chi_X$]{\epsfig{file=pics/molemmatchi.eps, width=6cm}}
    \caption{Graph of $\chi_X$ (dotted line) and approximate graphs of
      (a) $\Lambda \chi_X$, (b) $R_1 * \chi_X$ and 
      (c) $L \chi_X = \alpha(\Lambda\chi_X + \mu R_1 * \chi_X)$ 
      (solid line) for $X = B^2 \subset \R^2$;
      section along a straight line through the origin.}
    \label{fig:lambdachi}
  \end{center}
\end{figure}
}

These features of $\Lambda f$ cause small details of low contrast 
to be highlighted. This is often useful to some extent,
but it would normally be expedient to be able to alleviate the cup effect.
A heuristic method has been proposed for this purpose:
adding to $\Lambda f$ a multiple of $R_1 * \chi_X$, which can also be 
computed locally as is evident from \equ{IntSmDaftEqualToR1ConvF}.
It turns out that $R_1 * \chi_X$ is continuous everywhere 
and analytic in $\Rn \setminus \partial X$, and that it behaves like 
$C + \dist(x, \partial X)$ inside $X$ and
like $(C + \dist(x, \partial X))^{1-n}$ outside $X$;
see Figure~\ref{fig:lambdachi}(b).
This linear combination $Lf := \alpha(\Lambda f + \mu R_1 * f)$,
where $\alpha$ and $\mu$ are constants, is not an approximation of $f$, 
but its qualitative behaviour is quite similar,
and this reconstruction has proved to be useful in practice. 
See Figure~\ref{fig:lambdachi}(c).
\cite{faridani92:local,faridani92:localexamples,faridani97:local2}

The value of $\mu$ that yields the most informative images must be
determined empirically for each picture.
The graph of $\Lambda \chi_X$ is on average steeper for smaller sets $X$,
which implies a stronger need for cup correction.
Therefore $\mu$ should be large if the principal
interest is in small details, and small if $Lf$ is to be nearly flat 
in large areas of constant density.
A rule of thumb of $\mu = c r_0^{-2}$, where $c \approx 6$ is an experimental
constant, has been devised for making $L \chi_{B(0,r_0)}$ nearly constant
in most of $B(0,r_0)$.
\cite{faridani97:local2}

Although the difference in sign of $\Lambda \chi_X$ inside and outside $X$
permits qualitative inference of discontinuities 
in the x-ray attenuation density, quantitative estimates for
the sizes of the jumps in $f$ cannot be directly made 
from the jumps in $\Lambda f$. 
Procedures for calculating the jumps in $f$ have, however, been developed. 
They first require dividing the object into subsets $X_j$
in which $f$ is assumed to be constant.
\cite{faridani97:local2,katsevich95:jumps}

Observe that the reconstruction of $\Lambda f$ (or $\Lambda e * f$)
is called \emph{local} because the convolution kernel $\laplace \PP_\theta e$ 
in the reconstruction formula \equ{EConvLambdaFReconstructionFormula}
is compactly supported, so that
the computation of $\Lambda f(x)$ only requires measurements
of the attenuation along lines passing close to $x$.
On the other hand, the result is \emph{global} in the sense that 
the value of $f$ at a point $x$ affects $\Lambda f$ everywhere 
since $\Lambda$ is a non-local operator,
as mentioned in Section~\ref{sec:LocalTomography}.
In the reconstruction of $f$ (or $e * f$) 
the situation is reversed.
The kernel $\Lambda \PP_\theta e$ is not compactly supported,
whence the reconstruction of $f$ is \emph{global}, 
using attenuation measurements along all lines going through the object, 
but the result is \emph{local}:
changing the value of $f$ at a point $x$ does not change the 
reconstructed value at other points, except through the convolution
with the point spread function $e$.

This local-global duality is reflected in the fact that
$\Lambda \chi_X$ does not vanish completely outside $X$.
In particular, the high contrast details show in other parts of the 
locally reconstructed image.
Methods for reducing this effect have been developed.
\cite{faridani97:local2}

Several extensions to the concept of local tomography have also been proposed.
\emph{Pseudolocal tomography}, for instance, expresses the reconstructed
function as $f = f_d + f_d^c$, where $f_d^c$ is smooth.
The points of discontinuity and jumps in $f_d$ and its derivatives are exactly 
the same as those in $f$ and its derivatives. 
The reconstruction of $f_d$ can be done using measurements 
from only a small neighbourhood of the region of interest. 
\cite{katsevich96:pseudolocal,ramm96:radon_and_lt}

Other generalisations include 
considering different pseudodifferential operators $\mathcal{B}$ of $f$ 
than just $\Lambda f$. Under certain assumptions,
$\mathcal{B}$ preserves the singularities of $f$ as was shown in
Section~\ref{sec:WFLambdaFEqWFF} for $\Lambda$.
\cite{kuchment95:local,ramm96:radon_and_lt}

The theory can also be extended to the attenuated Radon transform,
which appears in the context of nuclear emission tomography.
Incomplete data problems, where measurements can only be made from
a limited angle, are also of both theoretical and practical interest.
\cite{kuchment95:local,natterer86,ramm96:radon_and_lt}

Any of these questions would make interesting subjects for further study.




\appendix 

\chapter{Mathematical Tools}
\label{app:MathematicalTools}


\section{Spaces of Functions and Distributions}
\label{sec:FunctionAndDistributionSpaces}


\subsection{$L^p$ Spaces}
\label{sec:Lp}

We state below some of the most often used features of $L^p$ spaces.
For an introduction to the theory of $L^p$ spaces, see \cite{rudin2}.
We shall give proofs or references to them
only for those claims that cannot be found in it.
%[Chapter 4 and Sections 6.15--6.17, 8.1--8.8]

In what follows, we shall denote the space $L^p(X)$, 
where $1 \le p \le \infty$ and $X$ is a measure space, simply by $L^p$. 
In this work, $X$ is always $\Rn$ or a measurable subset of it,
equipped with the Lebesgue measure.


For all $1 \le p \le \infty$, the spaces $L^p$ are complete normed spaces, 
\emph{i.e.} \emph{Banach spaces}, with norm $\nl{\cdot}{p}$. 

%The fact that $\nl{\cdot}{p}$ satisfies the triangle inequality, 
%which is required for a norm, is asserted by
%\begin{theorem}[Minkowski's inequality] \label{thm:Minkowski}
%  Let $1 \le p \le \infty$. Then
%  \begin{equation}\nt
%    \nl{f+g}{p} \le \nl{f}{p} + \nl{g}{p}.
%  \ent\end{equation}
%\end{theorem}

Convergence in $L^p$ ($\nl{f_k - f}{p} \to 0$) and pointwise convergence 
($f_k(x) \to f(x)$) do not imply each other, but the following is true: 
If a sequence of functions $(f_k)$ converges to $f$ in $L^p$,
it has a subsequence that converges to $f$ almost everywhere.

If $1 \le p, p' \le \infty$ are such that
\begin{equation}\nt
  \frac{1}{p} + \frac{1}{p'} = 1,
\ent\end{equation}
then $p$ and $p'$ are called \emph{conjugate exponents};
here $1/\infty = 0$. \label{def:ConjExp}
The notation $p'$ is often used without separate mention
for the conjugate exponent of $p$.
The spaces $L^p$ and $L^{p'}$ are closely related, as the following
results show.

\begin{theorem}[Hölder's inequality] \label{thm:Holder}
  Let $p$ and $p'$ be conjugate exponents. Then
  \begin{equation}\nt
    \nl{fg}{1} \le \nl{f}{p} \nl{g}{p'}.
  \ent\end{equation}
\end{theorem}

\begin{corollary} \label{cor:HolderForNFunctions}
  Let $0 \le p_1, p_2, \ldots, p_n \le \infty$ be such that
  \begin{equation}\nt
    \frac{1}{p_1} + \frac{1}{p_2} + \cdots + \frac{1}{p_n} = 1.
  \ent\end{equation}
  Then
  \begin{equation}\nt
    \nl{f_1 f_2 \cdots f_n}{1} 
    \le \nl{f_1}{p_1} \nl{f_2}{p_2} \cdots \nl{f_n}{p_n}.
  \ent\end{equation}
\end{corollary}
\begin{tod}
  We proceed by induction. For $n = 1$, the claim is trivial.
  Assume that the claim holds for $n=k$. Writing 
  $\frac{1}{p} = \frac{1}{p_1} + \frac{1}{p_2} + \cdots + \frac{1}{p_k}$
  so that $\frac{1}{p} + \frac{1}{p_{k+1}} = 1$,
  Hölder's inequality gives us
  \begin{equation}\nt
  \begin{split}
    \nl{f_1 \cdots f_k f_{k+1}}{1}
    & 
    \le \left( \int |f_1 \cdots f_k|^p\,dx \right)^{1/p} \nl{f_{k+1}}{p_{k+1}}
    \\ &
    = \left( \nl{f_1^p \cdots f_k^p}{1} \right)^{1/p} \nl{f_{k+1}}{p_{k+1}}.
  \end{split}
  \ent\end{equation}
  Now, since $\frac{1}{p_1/p} + \frac{1}{p_2/p} + \cdots +\frac{1}{p_k/p} = 1$,
  we can use our assumption:
  \begin{equation}\nt
  \begin{split}
    \nl{f_1^p \cdots f_k^p}{1}
    & \le \left( \int |f_1^p|^{p_1/p}\,dx \right)^{p/p_1} \cdots 
          \left( \int |f_k^p|^{p_k/p}\,dx \right)^{p/p_k}
    \\ &
    = \nl{f_1}{p_1}^p \cdots \nl{f_k}{p_k}^p,
  \end{split}
  \ent\end{equation}
  and hence
  \begin{equation}\nt
    \nl{f_1 \cdots f_k f_{k+1}}{1}
    \le \nl{f_1}{p_1} \cdots \nl{f_k}{p_k} \nl{f_{k+1}}{p_{k+1}}.
  \ent\end{equation}
  Thus, the claim holds equally well for $n = k+1$.
\end{tod}

If $1 \le p < \infty$ and $p$ and $q$ are conjugate exponents,
then $L^q$ is isometrically isomorphic to the \emph{dual space} of $L^p$.
This means that each bounded linear functional $\Phi: L^p \to \C$
can be represented by a unique $g \in L^q$ in the sense that 
for all $f \in L^p$,
\begin{equation}
  \label{LpLpPrimeDuality}
  \Phi(f) = \int f(x) \, \conj{g(x)} \,dx.
\end{equation}
This also implies
\begin{theorem}[Converse of Hölder's inequality]
  Let $p$ and $p'$ be conjugate exponents, $1 \le p < \infty$. Then
  \begin{equation}\nt
    \nl{f}{p} 
    = \sup_{\nl{g}{p'} \le 1} \Big| \int f(x) \, \conj{g(x)} \,dx \, \Big|.
  \ent\end{equation}
\end{theorem}

We shall often need to change orders of integration, which is justified by
\begin{theorem}[Fubini] \label{thm:Fubini}
  Let either
  \begin{enumerate}
    \vspace{-2ex}
    \item[(a)] $f: X_1 \times X_2 \to [0, \infty]$, or
    \item[(b)] $f: X_1 \times X_2 \to \C$ and 
      $f \in L^1(X_1 \times X_2)$.
    \vspace{-2ex}
  \end{enumerate}
  Then
  \begin{equation}\nt
    \int_{X_1} \left( \int_{X_2} f(x,y) \,dy \right) \,dx =
    \int_{X_2} \left( \int_{X_1} f(x,y) \,dx \right) \,dy.
  \ent\end{equation}
\end{theorem}

The \emph{convolution} \label{def:convolution} 
of two sufficiently well-behaved functions $f$ and $g$ is the function
\begin{equation}\nt
  (f*g)(x) = \int f(x-y) g(y) \,dy.
\ent\end{equation} 
Convolution has many useful properties, being for instance
commutative ($f*g = g*f$), associative ($f*(g*h) = (f*g)*h$), 
and linear ($f*(g+h) = f*g + f*h$). 
Also, 
%$
%  \smash{\dodo{}{x_j} (f*g)} = \smash{\dodo{f}{x_j} * g} 
%  = \smash{f * \dodo{g}{x_j}},
%$
\begin{equation}
  \label{DerivativeOfConvolutionForSmoothFunctions}
  \dodo{}{x_j}\, (f*g) = \dodo{f}{x_j} * g = f * \dodo{g}{x_j},
\end{equation}
when $f$ and $g$ are sufficiently smooth
(see Section \sectref{def:DistributionConvolution}).
Continuity is asserted by
\begin{theorem}[Young's inequality] \label{thm:Young}
  Let $p$, $q$ and $r$ be such that 
  $1 \le p, q, r \le \infty$ and 
  $\frac{1}{p} + \frac{1}{q} - \frac{1}{r} = 1$.
  Then
  \begin{equation}\nt
    \nl{f * g}{r} \le \nl{f}{p} \nl{g}{q}.
  \ent\end{equation}
\end{theorem}
\begin{tod}
%  \cite{neri71:singular}
  Since 
  \begin{equation}
    |f * g(x)|
    = \Big| \int f(x - y) g(y) \,dy \, \Big|
    \le \int |f(x - y)| |g(y)| \,dy
    = |f| * |g|(x),
  \end{equation}
  assume that $f$ and $g$ are real-valued and nonnegative; if they are not,
  set $\tilde{f} = |f|$, $\tilde{g} = |g|$ and drop the tildes.
  Thus also $(f*g)(x) \ge 0$ and $|f*g| = f*g$. 
  Let $1 \le \mu, \nu \le \infty$ be such that
  \begin{equation}\nt
    \frac{1}{\mu} = \frac{1}{p} - \frac{1}{r}, \qquad
    \frac{1}{\nu} = \frac{1}{q} - \frac{1}{r},
  \ent\end{equation}
  so that $\frac{1}{r} + \frac{1}{\mu} + \frac{1}{\nu} = 
  \frac{1}{p} + \frac{1}{q} - \frac{1}{r} = 1$. Thus, using
  the Corollary~\ref{cor:HolderForNFunctions} of Hölder's inequality, we have
  \begin{multline}\nt
    \begin{split}
      (f*g) (x) &
      = \int f(y)^{p(\frac{1}{p}-\frac{1}{\mu})} 
             g(x-y)^{q(\frac{1}{q} - \frac{1}{\nu})}
             f(y)^{p/\mu} g(x-y)^{q/\nu} \,dy \\ &
      \le \left(\int f(y)^{r p(\frac{1}{p}-\frac{1}{\mu})}
                     g(x-y)^{r q(\frac{1}{q}-\frac{1}{\nu})}
                                            \, dy \right)^{\frac{1}{r}}
    \, \times 
    \end{split}
    \\ \times \,
      \left(\int f(y)^p\,dy\right)^{\frac{1}{\mu}}
      \left(\int g(x-y)^q\,dy\right)^{\frac{1}{\nu}}.
  \ent\end{multline}
  Since
  \begin{equation}\nt
    \frac{1}{p} - \frac{1}{\mu} = \frac{1}{q} - \frac{1}{\nu} = \frac{1}{r},
  \ent\end{equation}
  this simplifies to
  \begin{equation}\nt
    (f*g) (x) \le
      \nl{f}{p}^{p/\mu} \nl{g}{q}^{q/\nu}
        \left(\int f(y)^p g(x-y)^q \, dy \right)^{\frac{1}{r}}
  \ent\end{equation}
  yielding
  \begin{equation}\nt
  \begin{split}
    \nl{f*g}{r}^r & = \int |(f*g)(x)|^r \,dx
    \\ &
    \le \nl{f}{p}^{pr/\mu} \nl{g}{q}^{qr/\nu}
        \int \int f(y)^p g(x-y)^q \,dy \,dx
    \\ &
    = \nl{f}{p}^{pr/\mu} \nl{g}{q}^{qr/\nu}
        \int f(y)^p \int g(x-y)^q \,dx \,dy
    \\ &
    = \nl{f}{p}^{pr/\mu} \nl{g}{q}^{qr/\nu} \nl{f}{p}^p \nl{g}{q}^q
  \end{split}
  \ent\end{equation}
  where reversing the order of integration is justified by Fubini's theorem
  (Theorem~\ref{thm:Fubini}) since $f^p$ and $g^q$ are nonnegative. 
  Thus, noting that
  \begin{equation}\nt
    \frac{1}{\mu} + \frac{1}{r} = \frac{1}{p} \qquad\text{and}\qquad
    \frac{1}{\nu} + \frac{1}{r} = \frac{1}{q},
  \ent\end{equation}
  we get
  \begin{equation}\nt
    \nl{f*g}{r} \le \nl{f}{p}^{\frac{p}{\mu}+\frac{p}{r}} 
                    \nl{g}{q}^{\frac{q}{\nu}+\frac{q}{r}}
    = \nl{f}{p} \nl{g}{q}.
  \ent\end{equation}
\end{tod}

In general, $L^p$ functions are of course not continuous. Since
$\nl{f - g}{p} = 0$ if $f$ and $g$ disagree on a set of measure zero,
one cannot even speak of point values of an $L^p$ function.
Instead, elements of $L^p$ must be interpreted as equivalence
classes of functions agreeing almost everywhere.

Continuous, and what is more, smooth functions are, however, 
dense in $L^p$ for $1 \le p < \infty$:
If the non-negative functions $\phi_j \in C^k(\Rn)$, 
$j \in \N \cup \{\infty\}$, 
are such that $\nl{\phi_j}{1} = 1$ for all $j$ and
\begin{equation}
  \int_{|x| \ge \delta} \phi_j(x) \,dx \to 0
  \qquad \text{as } j \to \infty,
\end{equation}
then the sequence of functions $\phi_j * f$ converges to $f$ in the 
$L^p$ norm;\footnote{%
  Reference \cite[Theorem 9.10]{rudin2} covers this 
  with the particular choice of
  \begin{equation}
    \phi_j(x) = \sqrt{\frac{2}{\pi}} \, \frac{j}{1 + j^2 x^2}.
  \end{equation}
  The more general form can be found in 
  \cite[Theorem 21.37]{hewitt-stromberg:real_and_abstract}.
  }
the fact that they are in $L^p$ is clear from Young's inequality above.
The functions $\phi_j * f$ are in $C^k$ because
$\partial^\alpha (\phi_j * f) = (\partial^\alpha \phi_j) * f$.

The kernels $\phi_j$, called mollifiers, 
can be chosen to be, for instance, the functions
$e_{1/j}$ introduced below in \equ{def:ApproxDelta} as approximations
of the delta distribution. In that case, $e_{1/j} * f \in C^\infty$.
\label{sec:LpMollification}


Of the $L^2$ spaces, $L^2$ plays a special role.
As $p = 2$ is its own conjugate exponent, 
$L^2$ can be thought of as its own dual space;
this is expressed by saying that $L^2$ is \emph{reflexive}.
Since $L^2$ is also separable, which is to say that it has a dense countable 
subset, the Banach-Alaoglu theorem can be applied:

\begin{theorem}[Banach-Alaoglu]
  \label{thm:BanachAlaoglu}
  The closed unit ball in a separable reflexive Banach space $\mathcal{B}$
  is weakly sequentially compact.
  That is, every bounded sequence $(x_j)_{j \in \N}$ in $\mathcal{B}$
  has a weakly convergent subsequence $(x_{j_k})_{k \in \N}$
  whose weak limit $x$ satisfies
  \begin{equation}
    \norm{x} \le \sup_{j \in \N} \norm{x_j}.
  \end{equation}
\end{theorem}
\begin{tod}
  See \cite[Theorem 6.3.7]{hutson-pym80} or \cite[Theorem 3.15]{rudin3}.
\end{tod}

Another remarkable property of $L^2$ is that for $p = 2$,
the duality in \equ{LpLpPrimeDuality} defines an inner product through
\begin{equation}\nt
  \ip{f}{g} = \int f(x)  \, \conj{g(x)} \,dx,
\ent\end{equation}
This makes $L^2$ into a \emph{Hilbert space}, \ie,
a Banach space whose norm is induced from the inner product:
\begin{equation}
  \nl{f}{2} = \sqrt{\ip{f}{f}}.
\end{equation}


\subsection{Principal Value Integrals}

If $f = u^+ - u^- + i(v^+ - v^-)$, where $u^+$, $u^-$, $v^+$ and $v^-$
are nonnegative, measurable functions, the Lebesgue integral of $f$
is defined to be
\begin{equation}
  \label{LebesgueIntegralAsParts}
  \int_E f \, dx = \int_E u^+ \,dx - \int_E u^- \,dx
    + i \int_E v^+ \,dx - i \int_E v^- \,dx,
\end{equation}
whenever all terms on the right hand side are finite.
If they are not, and this is caused by a singularity at one point,
say the origin, it may happen that the positive and negative
parts of the function cancel out if the singularity is approached
at an equal speed from all directions, \ie\, the limit
\begin{equation}
  \label{pv}
  \pv \int f(x) \,dx
  =  \lim_{\eps\to 0^+} \int_{E \setminus B(0,\eps)} f(x) \,dx
\end{equation}
may be finite. 
In this case, we call this limit the \emph{Cauchy principal value}
of the integral.
\label{def:pv}

Convolution being an integral, too, we define the
\emph{Cauchy principal value convolution} of $f$ and $g$ as
\begin{equation}
  \label{pvConvolution}
  \pv f * g (x) = \pv \int f(y) g(x - y) \,dy.
\end{equation}
Note that the definition is not symmetric with respect to $f$ and $g$.

The letters $\pv$ are also sometimes used as part of 
the name of a distribution; this is the case when 
the principal value integral defines a distribution through
\begin{equation}
  \label{pvDistribution}
  \du{\pv f}{\phi} = \pv \int f(x) \, \phi(x) \,dx.
\end{equation}
One such situation is given by Theorem~\ref{thm:VPKIsADistribution}.
Distributions are introduced in the following section.




\subsection{Distributions}
\label{sec:Distributions}

For an open set $X \subset \Rn$, we shall call 
$\DD(X)$ the space $C_0^\infty(X)$ with a topology such that
a sequence $(\phi_k)$ converges $to \phi$ in $\DD(X)$, if
\begin{enumerate}
\samepage
  \item there exists a fixed compact set $K \subset X$ such that
    $\supp \phi_k \subset K$ for all $k \in \N$
  \item for all $m \in \N$, 
    $\sup \big\{| \partial^\alpha \phi_k(x) - \partial^\alpha \phi(x)| 
       \,\big|\, |\alpha| \le m, \, x \in K \big\} \xrightarrow{k \to \infty} 0$.
\end{enumerate}
\label{def:TestFunctions}

Distributions are defined as generalisations of functions in the following way.
For $\phi, \psi \in \DD(X)$, the bilinear pairing
\begin{equation}
  \label{PairingForTestFunctions}
  \du{\psi}{\phi} = \int_X \psi(x)\, \phi(x) \,dx
\end{equation}
makes $\du{\psi}{\cdot}\!\!: \DD(X) \to \C$ 
a continuous linear functional, as the integral is absolutely convergent.
More generally, we shall call \emph{distributions} on $X$ 
the continuous linear functionals on $\DD(X)$,
and denote by $\DD'(X)$ their complex vector space.
The value of a distribution $f$ at $\phi \in \DD(X)$ can be written 
\begin{equation}
  \label{PairingForDistributions}
  f(\phi) = \du{f}{\phi}.
\end{equation}
The latter form is normally used.
Convergence in $\DD'(X)$ is defined as follows: $f_k \to f$ if
$\du{f_k}{\phi} \to \du{f}{\phi}$ for all $\phi \in \DD(X)$. 
For these reasons, the elements of $\DD(X)$ 
are called \emph{test functions}.
\label{def:Distributions}


According to Equation~\equ{PairingForTestFunctions}, 
every test function $\psi \in \DD(X)$ can also be considered 
a distribution. 
In the sequel, we shall often make no distinction between a test function
and the corresponding distribution, and we shall even write\footnote{%
  In fact, this should be interpreted as
  $\iota: \DD(X) \to \DD'(X)$, where
  $\du{\iota\psi}{\phi} = \int_X \psi(x)\,\phi(x) \,dx$.
  The inclusions in Figure~\ref{fig:FunctionSpaces} should be understood 
  similarly.
}
\begin{equation}\nt
  \DD(X) \subset \DD'(X).
\ent\end{equation}
Consequently, when we extend operations on functions to operations on 
distributions, we shall have to make sure that when restricted to 
$\DD(X)$, they agree with the familiar definitions.

Figure~\ref{fig:FunctionSpaces} shows some inclusion relations
between spaces of functions and distributions used in this work.

\begin{figure}[htbp]
  \begin{center}
      \xypoisto{\input{pics/fcn-spaces}}
    \caption{Inclusion relations between various function spaces.}
    \label{fig:FunctionSpaces}
  \end{center}
\end{figure}

A particularly useful class of test functions are the
approximate characteristic functions of balls $B(0,\eps_1)$,
that is, functions that assume the value 1 in $B(0,\eps_1)$ 
and the value 0 outside larger balls $B(0,\eps_2)$.
Such functions are given, for instance, by the formula
\begin{equation}
  \label{ApproxCharacteristicFunction}
  \tilde{\chi}_{\eps_1, \eps_2} (x) = 
  \begin{cases}
    1, & |x| \le \eps_1 \\
    \exp\left( \frac{1}{(\eps_2-\eps_1)^2} 
               + \frac{1}{(|x|-\eps_1)^2-(\eps_2-\eps_1)^2} \right),
       & \eps_1 \le |x| \le \eps_2 \\
    0, & |x| \ge \eps_2.
  \end{cases}
\end{equation}
(See Figure~\ref{fig:approx-chi}.) \label{def:ApproxCharacteristicFunction}
\kuvienpoisto{\begin{figure}[htbp]
  \begin{center}
    \leavevmode
\epsfig{file=pics/approx-chi.eps,width=10cm,bbllx=72,bblly=180,bburx=540,bbury=580}
    \caption{The approximate characteristic function
      $\tilde{\chi}_{\eps_1, \eps_2}$
      in \equ{ApproxCharacteristicFunction} with
      $n = 2$, $\eps_1 = 0.3, \, \eps_2 = 1.4$.}
    \label{fig:approx-chi}
  \end{center}
\end{figure}}

The \emph{Schwartz space of rapidly decreasing functions} is the set
\begin{equation}\nt
  \Ss(X) 
  = \big\{ \phi \in C^\infty(X) \,\big|\, 
      \norm[\alpha, \beta]{\phi} < \infty 
      \;\, \forall\, \alpha, \beta \in \N^n \big\},
\ent\end{equation}
where the numbers
\begin{equation}\nt
  \norm[\alpha, \beta]{\phi} 
  = \sup_{x\in X} |x^\alpha \,\partial^\beta \phi(x)|
\ent\end{equation}
are seminorms defining the topology of $\Ss$. 
The space $\Ss(X)$ is metrisable, but for our purposes, 
it suffices to know that $\phi_k \to \phi$ in $\Ss$ if and only if
$\norm[\alpha,\beta]{\phi_k - \phi} \to 0$ for all $\alpha, \beta \in \N^n$.
\label{def:Ss} 

A simple consequence of the finiteness of the seminorms 
$\norm[\alpha,\beta]{\phi}$
%is that $p(x, |x|) \phi(x) \in L^\infty(X) \cap L^1(X)$
is that $p(x, |x|) \phi(x)$ is bounded and integrable
for all polynomials $p$ of the $n + 1$ variables 
$x_1,\ \ldots,\, x_n,\, |x|$.

Analogously to $\DD'(X)$, we define $\Ss'(X)$,
the space of \emph{tempered distributions} on $X$, to be the set of
continuous linear functionals on $\Ss(X)$. 
As before in \equ{PairingForDistributions}, 
the notation $\du{f}{\phi}$ will be used for the value that 
a tempered distribution $f$ assumes at $\phi \in \Ss(X)$.
An equivalent definition for $\Ss'$ is
\begin{equation}\nt
  \Ss'(X) = \big\{ f: \Ss(X) \to \C \,\big|\, \exists\, C \ge 0,\, N\in\N: \;
    |\du{f}{\phi}| 
    \le C \sum_{\substack{|\alpha| \le N \\ |\beta| \le N}}
          \norm[\alpha,\beta]{\phi} 
    \;\,\forall\, \phi\in\Ss \big\}.
\ent\end{equation}
A sequence $(f_k)$ in $\Ss'(X)$ converges to $f$ 
if $\du{f_k}{\phi} \to \du{f}{\phi}$ for all $\phi \in \Ss(X)$.
This weak topology of $\Ss'(X)$ is not metrisable.
\label{def:Ssp}
As arguments of tempered distributions, rapidly decreasing functions
are also called test functions.

Again, $\Ss(X)$ can be considered a subset of $\Ss'(X)$
if we are careful with extending operations on rapidly decreasing
functions to operations on tempered distributions.
The fact that this inclusion is dense is remarkable, because
it allows us to extend many operators $L: \Ss(X) \to \Ss(X)$
to $\tilde{L}: \Ss'(X) \to \Ss'(X)$ in a sequentially continuous way, 
\ie, in such a way that if
$\psi_k \to f \in \Ss'(X)$ with $\psi_k \in \Ss(X)$, then
\begin{equation}\nt
  \du{L \psi_k}{\phi} \xrightarrow{k \to \infty} \du{\tilde{L} f}{\phi}
\ent\end{equation}
for all $\phi \in \Ss(X)$.
Definitions of sequentially continuous extensions include
multiplication by a function $M \in C^\infty$ whose all derivatives
are dominated in absolute value by a polynomial,
\begin{equation}\nt
  \du{Mf}{\phi} = \du{f}{M \phi},
\ent\end{equation}
and differentiation: The \emph{distribution derivative} 
or \emph{weak derivative} $D^\alpha$
is defined by \label{def:DistributionDerivative}
\begin{equation}\nt
  \du{D^\alpha f}{\phi} = \du{f}{(-\partial)^\alpha \phi},
\ent\end{equation}
whose consistency can be seen by partial integration.
For first order distribution derivatives, the notation $D_j := D^{e_j}$
is also used.

The \emph{dilation operator} $\sigma_\lambda$, $\lambda > 0$, is defined for
a function $\phi \in \Ss$ by $\sigma_\lambda \phi(x) = \phi(\lambda x)$.
For $f \in \Ss'$, we set
\begin{equation}\nt
  \du{\sigma_\lambda f}{\phi} = \lambda^{-n} \du{f}{\sigma_{1/\lambda}\phi}.
\ent\end{equation}
This, too, is a natural extension to the definition above, 
because for $\psi \in \Ss$,
the change of variable $y = x/\lambda$, $dy = \lambda^{-n} dx$ yields
\begin{equation}\nt
  \lambda^{-n} \du{\psi}{\sigma_{1/\lambda}\phi}
  = \int \psi(x) \, \phi({\tfrac{x}{\lambda}}) \, \lambda^{-n} \,dx
  = \int \psi(\lambda y) \, \phi(y) \,dy = \du{\sigma_\lambda \psi}{y}.
\ent\end{equation}
\label{def:dilation}

The \emph{support} of a function $f$ defined on $X$ 
is the set \label{def:supp}
\begin{equation}\nt
  \supp f = \overline{\{ x \in X \,|\, f(x) \neq 0 \}};
\ent\end{equation}
$\overline{A}$ denotes the \emph{closure} of a set $A$,
\ie, the smallest closed set containing $A$. \label{def:closure}

A distribution $f \in \DD'(X)$ is said to 
vanish on a set $U \subset X$, if $\du{f}{\phi} = 0$
for all $\phi$ with $\supp \phi \subset U$.
The \emph{support of a distribution} $f \in \DD'(X)$ is
defined as the set of points that do not have a neighbourhood 
on which $f$ vanishes.
%\begin{equation}\nt
%  \supp f = X \setminus \{ x \in X \,|\, 
%  \exists\, U \subset X, \; U \text{ open},\, x \in U :\: 
%  \forall\,\phi \in \DD(X):\;
%  \supp \phi \subset U \!\!\imp\!\! \du{f}{\phi} = 0 \}.
%\ent\end{equation}

The space of distributions with compact support
is denoted by $\mathcal{E}'(X) = \DD'_0(X)$.
These distributions can be extended to linear functionals on 
$C^\infty(X) =: \mathcal{E}$, since for $f \in \DD_0'$ with 
$\supp f \subset B(0,R)$ and $\phi \in C^\infty$,
\begin{equation}
  \du{f}{\phi}
  = \du{f}{\chim_{R,R+1} \phi}
\end{equation}
is finite as $\chim_{R,R+1} \phi \in C^\infty_0$.
The same topology is used as for $\DD'(X)$.
\label{def:DistributionsWithCompactSupport}

Convolutions $f * g$ can be defined for $f \in \DD'(X)$
and $g \in \DD(X)$ by
\begin{equation}\nt
  \du{f * g}{\phi} = \du{f}{\refl{g} * \phi},
\ent\end{equation}
where $\refl{g}$ denotes the \emph{reflection} of $g$, 
$\refl{g}(x) = g(-x)$. \label{def:reflection} 
This is the natural extension from the case where $f \in \Lloc$.
The same formula can then be used to extend the definition to cases
where $g \in \DD'_0(X)$. \label{def:DistributionConvolution}
Most properties of the convolution, including commutativity, linearity,
the Convolution theorem \equ{ConvolutionTheorem} and
Formula~\equ{DerivativeOfConvolutionForSmoothFunctions} for the derivatives,
remain valid after this extension.

Inclusion relations between these function and distribution spaces 
are shown in Figure~\ref{fig:FunctionSpaces}.
An inclusion $A \subset B$ is denoted by $A \to B$.
All inclusions are proper.

An example of a distribution which is not a conventional function
is \emph{Dirac's delta distribution} $\delta$, defined by
\begin{equation}\nt
  \du{\delta}{\phi} = \phi(0).
\ent\end{equation}
Clearly, there is no function $e: X \to \C$ such that
\begin{equation}\nt
  \int_X e(y) \phi(y) \,dy = \phi(0),
\ent\end{equation}
for all $\phi \in \DD$, but as $\DD$ is dense in $\Ss'$, 
$\delta$ can be seen as the limit of a sequence of functions having 
higher and higher, narrower and narrower peaks around the origin.
For instance, if we define 
\begin{equation}
  \label{def:ApproxDelta}
  e_\eps(x) = \eps^{-n} e_1(x/\eps), \quad\text{where}\quad
  e_1(x) = \chim_{0,1}(x) =
  \begin{cases}
    C e^{\frac{1}{|x|^2-1}}, & |x|  <  1 \\
    0,                       & |x| \ge 1
  \end{cases}
\end{equation}
and choose $C$ in such a way that $\int e_1(x) \,dx = 1$,
we have $e_\eps \in \Ss$ for all $\eps \ge 0$ and
\begin{equation}\nt
  \delta = \lim_{\eps \to 0} e_\eps.
\ent\end{equation}
Such a function $e_\eps$ is often called an \emph{approximate delta function},
\emph{blurring kernel} or \emph{point spread function}.

The derivatives of the delta distribution give
the values of the corresponding derivatives of the test function 
at the origin,
\begin{equation}
  \du{D^\alpha \delta}{\phi} 
  = (-1)^{|\alpha|} \du{\delta}{\partial^\alpha \phi}
  = (-1)^{|\alpha|}\, \partial^\alpha \phi(0).
\end{equation}
The support of the delta distribution and its derivatives 
consists of the single point $0$. 
In fact, finite linear combinations of $\delta$ and its derivatives  
are the only functions with point support:
if $\supp f = \{ 0 \}$, then 
$f = \sum_{|\alpha| \le m} c_\alpha D^\alpha \delta$ for some
constants $m$ and $c_\alpha$.


For an introduction to the theory distributions, see references
\cite{bremermann65:distr,friedlander82:distr,hormander,
  neri71:singular,rauch91:pde}.%
\footnote{\cite{bremermann65:distr} includes a gentle, 
  intuitive prologue.
  \cite{friedlander82:distr} provides a good treatment of the
  basic theory.
  An exhaustive account of the theory can be found in \cite{hormander}.

  Reference \cite{neri71:singular} also gives a short introduction
  to the theory of distributions. 
  Like some other texts, it defines distributions as the continuous 
  \emph{conjugate} linear functionals on $\DD$, which has the advantage 
  that $\du{f}{\phi} = \ip{f}{\phi}$ for $f \in L^2$, $\phi \in \Ss$, 
  but this changes somewhat most formulae. 
  Qualitatively, however, the results remain obviously the same, 
  because each test function is, in fact, only exchanged 
  with its complex conjugate.

  \cite{rauch91:pde} deals with $\DD'$ only briefly in an appendix,
  but considers more thoroughly tempered distributions, their 
  Fourier transforms and their relations with $L^p$ spaces.
}


\subsection{Fourier Transform}
\label{sec:FourierTransform}

Various different definitions are commonly used for the 
\emph{Fourier transform} $\F f=\Ff$ of a function
$f:\Rn \to \C$ in an appropriate class, for instance, $f \in \Ss$.
We shall use the following one:
\label{def:FourierTransform} \label{def:InverseFourierTransform}
\begin{equation}
  \label{FourierTransformForSs}
  \F f(\xi) = \Ff(\xi) = \Fm \int_\Rn f(x) \, e^{-i x \cdot \xi} \,dx.
\end{equation}
The \emph{inverse Fourier transform} is then given by
\begin{equation}
  \label{InverseFourierTransformForSs}
  f(x) = \Fi{\Ff}(x) 
  = \Fm \int_\Rn \Ff(\xi) \, e^{i x \cdot \xi} \,d\xi.
\end{equation}
The Fourier transform can be considered as a linear operator between
various function spaces.
Linearity is of major significance, as many operations considered
in this work are linear.

The Fourier transform $\mathcal{F}: \Ss \to \Ss$ 
is a continuous linear bijection. Also,
\begin{equation}\nt
  |\Ff(\xi)| \le \Fm \int |f(x)| \, |e^{-ix \cdot \xi}| \,dx
  = \Fm \nl{f}{1},
\ent\end{equation}
whence $\nl{\Ff}{\infty} \le \Fm\, \nl{f}{1}$, so that
the Fourier transform can be extended to a continuous linear injection from
$L^1$ to $L^\infty$, using the same Formula~\equ{FourierTransformForSs}.

The Fourier transform $\mathcal{F}: \Ss \to \Ss$ can also be extended
to a bijection $\mathcal{F}: L^2 \to L^2$
as the limit of the truncated Fourier transform:
\begin{equation}\nt
  \label{FourierTransformForLTwo}
  \F f(\xi) = \Ff(\xi)
  = \ltwolim_{R \to \infty}\, 
    \Fm \int_{|x|\le R} f(x) \, e^{-ix \cdot \xi} \, dx.
\ent\end{equation}
The inverse transform is then
\begin{equation}\nt
  \label{InverseFourierTransformForLTwo}
  f(x) = \ltwolim_{R \to \infty} 
    \Fm \int_{|x| \le R} \Ff(\xi) \, e^{ix \cdot \xi} \, d\xi.
\ent\end{equation}

The Fourier transform and its inverse are extended to members of $\Ss'$ by
\begin{equation}\nt
  \du{\F f}{\phi} = \du{f}{\F \phi} \qquad
  \du{\Fi f}{\phi} = \du{f}{\Fi \phi} \qquad \text{for}\; \phi \in \Ss.
\ent\end{equation}
This operator $\F: \Ss' \to \Ss'$ is sequentially continuous,
and its restrictions to $\Ss$, $L^1$ and $L^2$ agree with
the Fourier transforms defined in \equ{FourierTransformForSs}
and \equ{FourierTransformForLTwo}.

The letter $\xi$ will commonly be used without separate mention
for the variable on the Fourier transform side, whereas
$x$ will be used for the variable on the untransformed side.

Some additional properties of the Fourier transform include
the following:
\begin{alignat}{2}
  \F(\partial^\alpha f)(\xi) &= (i\xi)^\alpha \Ff(\xi)
    & \qquad \text{for } & f \in \Ss'
    \label{FourierTransformOfDerivatives} \\
  \F(\sigma_\lambda f) &= \lambda^{-n}\sigma_{1/\lambda} \Ff 
    & \qquad \text{for } & f \in \Ss',
    \label{FourierTransformOfDilation} \\
  \F\F f(x) = \Fi\Fi f(x) &= \refl{f}(x) := f(-x)
    & \qquad \text{for } & f \in \Ss',
    \label{FourierTransformAntiInvolutivity} \\
\intertext{the \emph{Convolution theorem}}
  \F(f * g)(\xi) &= \Fp\, \Ff(\xi) \, \widehat{g}(\xi)
%   & \qquad \text{for } & f \in L^1,\, g \in L^1 \cup L^2 
    \label{ConvolutionTheorem} \\
\intertext{%
for $f \in L^1,\, g \in L^1 \cup L^2$ or $f \in \Ss',\, g \in \DD'_0$,
the \emph{Parseval formula}}
  \nl{\F f}{2} & = \nl{f}{2} 
    & \qquad \text{for } & f \in L^2,
  \label{PFormulaNorm} \\
\intertext{the \emph{Plancherel formula}\footnotemark}
%  \int f(x) \conj{g(x)} \, dx &= \int \Ff(\xi) \conj{\Fg(\xi)} \, d\xi
   \ip{f}{g} &= \ip{\Ff}{\Fg}
    & \qquad \text{for } & f, g \in L^2 
  \label{PFormulaIP} \\
  \du{f}{\conj{\phi}} &= \du{\Ff}{\conj{\widehat{\phi}}}
    & \qquad \text{for } & f \in \Ss',\, \phi \in \Ss
  \label{PFormulaDuality} \\
\intertext{and the Riemann-Lebesgue lemma}
  \lim_{|x| \to \infty} &\Ff(x) = 0
    & \qquad \text{ for } & f \in L^1.
  \label{RiemannLebesgue}
\end{alignat}
\footnotetext{
  The resemblance between the names Parseval and Plancherel causes some 
  inconsistency as to which name refers to which one of these
  two equivalent formulae.
  We choose to call the former one \equ{PFormulaNorm} the Parseval formula,
  and the latter one \equ{PFormulaIP}, \equ{PFormulaDuality} 
  the Plancherel formula.
}

As \equ{FourierTransformOfDilation} suggests, 
\begin{equation}
  \F(\delta) 
  = \lim_{\eps \to 0} \F(\eps^{-n} \sigma_{1/\eps} e)
  = \lim_{\eps \to 0} \sigma_\eps \widehat{e}
  = 1,
\end{equation}
when $e$ is an approximate delta function.

References \cite{hormander},
\cite{neri71:singular}, \cite[Chapter 2]{rauch91:pde},
and \cite[Chapter 9]{rudin2} provide an introduction to the 
Fourier transform, together with proofs of these facts.


\subsection{$H^s$ Spaces}
\label{sec:Hs}

For the Sobolev spaces $H^k(\Rn)$ with $k \in \N$, defined in 
Chapter~\sectref{def:Hs}, we note that an equivalent definition is
\begin{equation}\nt
  H^k(\Rn) 
  = \left\{ f \in L^2(\Rn) \,|\, D^\alpha f \in L^2(\Rn) 
      \text{ for all } \alpha \in \N^n \text{ with } |\alpha| \le k \right\}.
\ent\end{equation}
This is obvious because if $|\alpha| \le k$,
\begin{equation}\nt
  |\xi^\alpha| = \prod_{j=1}^n |\xi_j|^{\alpha_j}
  \le \prod_{j=1}^n |\xi|^{\alpha_j}
  = |\xi|^k
  \le (1 + |\xi|^2)^{k/2}
\ent\end{equation}
implies that
\begin{equation}\nt
  \nl{D^\alpha f}{2} = \nl{\F(D^\alpha f)}{2} = \nl{\xi^\alpha \Ff}{2}
  \le \nl{(1 + |\xi|^2)^{k/2}\, \Ff}{2},
\ent\end{equation}
and on the other hand,
\begin{equation}\nt
  (1 + |\xi|^2)^{k/2} 
  \le \Big(1 + \sum_{j=1}^n |\xi_j| \Big)^k
  = \sum_{|\alpha| \le k} c_\alpha \, |\xi^\alpha|
\ent\end{equation}
implies that
\begin{equation}
\begin{split}
  \nl{(1 + |\xi|^2)^{k/2} \,\Ff}{2}
  &
%  \le \Nl{\smash[b]{\sum_{|\alpha| \le k} c_\alpha |\xi^\alpha|\, \Ff}}{2}
  \le \sum_{|\alpha| \le k} |c_\alpha|\, \nl{|\xi^\alpha|\, \Ff}{2}
  \\&
  =   \sum_{|\alpha| \le k} |c_\alpha|\, \nl{\F(D^\alpha f)}{2}
  \\&
  =   \sum_{|\alpha| \le k} |c_\alpha|\, \nl{D^\alpha f}{2}
\end{split}
\end{equation}
so that $\nl{(1 + |\xi|^2)^{k/2}\, \Ff}{2} < \infty$
if and only if $\nl{D^\alpha f}{2} < \infty$ whenever $|\alpha| < k$.



\section{Continuity of the Riesz Potentials}
\label{sec:ContinuityOfRieszPotentials}

The definition of the Riesz potential, given in 
Definition~\ref{def:RieszPotential}, can be generalised:
The \emph{Riesz potential of order $\alpha \in \R$} of the function 
$f : \Rn \to \C$ is the function $R_\alpha * f$, where
\begin{equation}
  R_\alpha = \frac{\Gamma\big(\frac{n - \alpha}{2}\big)}{\pi^{n/2} 
                                 2^\alpha \Gamma\big(\frac{\alpha}{2}\big)}
             \, |x|^{\alpha - n}.
\end{equation}
Texts that use the notation $I_1$ instead of $R_1*$, 
also use $I_\alpha$ instead of $R_\alpha *$.

The mappings $f \mapsto R_\alpha * f$ have the following continuity properties,
which are needed in the proof of Theorem~\ref{thm:DerivativeOfRieszPotential}.
Instead of giving the proofs, we refer to \cite{stein70:sing} and
\cite{ziemer89:weakly_diff}.

\pagebreak
\begin{theorem}
  \label{thm:SteinThm1BPage119} \label{thm:ZiemerThm284}
  If $\alpha > 0$, $1 < p < \infty$ and $\alpha p < n$,
  then there is a constant $C$, depending on $n$ and $p$, such that
  \begin{equation}
    \nl{|x|^{\alpha - n} * f(x)}{q} \le C \, \nl{f}{p},
  \end{equation}
  where
  \begin{equation}
    q = \frac{np}{n - \alpha p}
  \end{equation}
  for all $f \in L^p(\Rn)$.
\end{theorem}

\begin{tod}
  See \cite[Theorem 2.8.4, page 86]{ziemer89:weakly_diff} or
  \cite[Theorem V.1(b), page 119]{stein70:sing}.
\end{tod}


\begin{theorem}
  \label{thm:ZiemerThm291}
  If $E \subset \Rn$ is compact and $p > 1$, 
  then there are constants $C_1$ and $C_2$, depending only on
  $p$, $n$ and $E$, such that
  \begin{equation}
    \frac{1}{m(E)} \int_E \exp \left[ \left( 
        \frac{|x|^{n/p - n} * f(x)}{C_1 \, \nl{f}{p}} \right)^{p'} \right] \,dx
    \le C_2
  \end{equation}
  for all $f \in L^p(E)$.
\end{theorem}

\begin{tod}
  See \cite[Theorem 2.9.1, page 89]{ziemer89:weakly_diff}.
\end{tod}


These two theorems imply that for all $n \ge 2$,
the Riesz potential operator is locally bounded 
from $L^2(\Rn)$ to some $L^q(\Rn)$.

\begin{corollary}
  \label{cor:ZiemerCorollary}
  If $n \ge 2$, and $E \subset \Rn$ is compact, then there
  are constants $q \ge 2$ and $C > 0$ such that 
  \begin{equation}
    \nl{\chi_E \, R_1 * f}{q} \le C \, \nl{f}{2}
  \end{equation}
  for all $f \in L^2(\Rn)$.
\end{corollary}

\begin{tod}
  For $n \ge 3$, the result follows directly from the global estimate of
  Theorem~\ref{thm:ZiemerThm284} with $\alpha = 1$ and $p = 2$.
  If $n = 2$, note that the expression $t e^{-t}$ is bounded 
  by some constant $C_3$ for all $t > 0$.
  Theorem~\ref{thm:ZiemerThm291} with $p = 2$ therefore implies that
  for any compact set $E \subset \Rn$,
  \begin{equation}
  \begin{split}
    \int_E \left|\frac{R_1 * f(x)}{C_1 \, \nl{\chi_E f}{2}}\right|^2
        \, dx
    &
    \le \int_E \left(\frac{R_1 * |f|(x)}{C_1 \, \nl{\chi_E f}{2}}\right)^2
        \, dx
    \\&
    \le C_3 \int_E \exp \left[\left(\frac{R_1 * |f|(x)}{C_1 \,
                                  \nl{\chi_E f}{2}}\right)^2 \right] \,dx
    \\&
    \le C_3 \, C_2 \, m(E)
  \end{split}
  \end{equation}
  and therefore
  \begin{equation}
  \label{ZiemerCorollaryNEQ2}
    \nl{\chi_E \, R_1 * f}{2}
    = \sqrt{\int_E |R_1 * f(x)|^2 \,dx}
    \le C \, \nl{\chi_E \, f}{2}
    \le C \, \nl{f}{2},
  \end{equation}
  where $C = C_1 \sqrt{C_3 \, C_2 \, m(E)}$.
\end{tod}

\pagebreak
\section{Polar and Spherical Coordinates in $\Rn$}
\label{sec:PolarCoordinates}

In $\Rn$, $n \ge 2$, an integral $\int_D f(x)\,dx$ can often be 
easily evaluated  using \emph{polar coordinates} 
if the integrand $f$ and domain of integration $D \subset \Rn$ 
possess appropriate symmetry.

If $x = (x_1, x_2, \ldots, x_n) \in \Rn$, the simplest form of 
polar coordinates is the well known
\begin{equation}\nt
  r = |x| \in [0, \infty), \qquad \theta = \frac{x}{r} \in \Sm, 
  \qquad   x = r \theta
\ent\end{equation}
so that
\begin{equation}
  \label{PolarCoordinates}
  \int_D f(x) \,dx 
  = \int_{D_\theta} \int_{D_r} f(r,\theta)\, r^{n-1} \,dr \,d\theta
\end{equation}
when
\begin{equation}
  D = \big\{ r\theta \in \Rn \,\big|\, 
      r \in D_r \subset [0,\infty),\, \theta \in D_\theta \subset \Sm \big\}.
\end{equation}
When $n = 1$, this holds equally well, if the counting measure is used 
on $D_\theta \subset S^0 = \{-1, +1\}$:
\begin{equation}
  \int_{D_\theta} f(\theta) \, d\theta
  = \sum_{\theta \in D_\theta} f(\theta).
\end{equation}
In this case, \equ{PolarCoordinates} simply becomes
\begin{equation}
  \label{PolarCoordinatesInR1}
  \int_D f(x) \,dx
  = \int_{(-\infty,0] \cap D} f(x) \,dx + \int_{[0,\infty) \cap D} f(x) \,dx.
\end{equation}
The same formalism can thus be used regardless of the dimension.
%Because of the triviality of \equ{PolarCoordinatesInR1},
We shall focus on the cases $n \ge 2$ for the rest of this section.

If $f = f(r)$ is spherically symmetric,
\begin{equation}\nt
  \int_D f(x) \,dx = m(D_\theta) \, \int_{D_r} f(r)\, r^{n-1} \,dr.
\ent\end{equation}
A simple application of this is the calculation of the measures of
the unit balls and spheres in different dimensions:

\begin{lemma}
\label{lemma:MeasureUnitSphereAndBall}
For all $n \in \Z_+$,
\begin{equation}
  \label{MeasureUnitSphereAndBallLemma}
  |S^{n-1}| = \frac{2\pi^{n/2}}{\Gamma\left(\frac{n}{2}\right)} = n\,|B^n|,
  \qquad\qquad
  |B^n| = \frac{2 \pi^{n/2}}{n \,\Gamma\left(\frac{n}{2}\right)}.
\end{equation}
\end{lemma}
\begin{tod}
  For calculating $|S^{n-1}|$, we shall evaluate the integral 
  \begin{equation}\nt
    I = \int_{\R^n} e^{-|x|^2}
  \ent\end{equation}
  in two ways. Firstly, since
  \begin{equation}\nt
  \begin{split}
    \infint e^{-x^2} \,dx &
    = \sqrt{\infint e^{-x^2}\,dx \infint e^{-y^2}\,dy}
    = \sqrt{\infint \infint e^{-(x^2+y^2)} \,dx\,dy} \\ &
    = \sqrt{\int_0^{2\pi} \int_0^\infty e^{-r^2}\,r\,dr\,d\theta}
    = \sqrt{2 \pi \sij{0}{\infty} -\frac{e^{-r^2}}{2}} = \sqrt{\pi},
  \end{split}
  \ent\end{equation}
  we get
  \begin{equation}\nt
    I = \left(\infint e^{-x^2} \,dx \right)^n = \pi^{n/2}.
  \ent\end{equation}
  Secondly,
  \begin{equation}\nt
    I = \int_{S^{n-1}} \int_0^\infty e^{-r^2} r^{n-1} \,dr \,d\theta,
  \ent\end{equation}
  whence, by making the substitution $u = r^2$, $du = 2r \,dr$, we get
  \begin{equation}\nt
    I = |S^{n-1}| \, \frac{1}{2} \, \int_0^\infty e^{-u} u^{\frac{n}{2}-1} \,du
    = \frac{1}{2} \,|S^{n-1}|\, \Gamma\left(\frac{n}{2}\right).
  \ent\end{equation}
  Therefore, 
  \begin{equation}\nt
    |S^{n-1}| = \frac{2 \pi^{n/2}}{\Gamma\left(\frac{n}{2}\right)},
  \ent\end{equation}
  proving the first part of the claim. The second part follows immediately
  by using polar coordinates:
  \begin{equation}\nt
    |B^n| = \int_{B^n} dx = \int_{S^{n-1}} \int_0^1 r^{n-1} \,dr\,d\theta
    = \frac{|S^{n-1}|}{n}.
  \ent\end{equation}
\end{tod}

Even when $f$ is not spherically symmetric, the evaluation of
\equ{PolarCoordinates} is rather straightforward in the two-dimensional case,
in the sense that both $D_r \subset [0, \infty)$ and $D_\theta \subset S^1$
are one-dimensional.

For $n \ge 3$, $D_\theta$ must be parametrised using 
$n - 1 \ge 2$ real numbers.
In some cases, this can be done by
introducing the \emph{spherical coordinates} 
$r \in D_r \subset [0, \infty)$, $\varphi \in D_\varphi \subset [0, \pi]$
and $\omega = (\omega_1, \omega_2, \ldots, \omega_{n-2}) 
\in D_\omega \subset B^{n-2}$:
\label{sec:SphericalCoordinates}
\begin{equation}
\begin{split}
\label{SphericalCoordinates}
  x_1 & = r \cos \varphi \\
  x_2 & = r \omega_1 \sin \varphi \\
  x_3 & = r \omega_2 \sin \varphi \\
  & \hspace*{1.2ex} \vdots \\
  x_{n-1} & = r \omega_{n-2} \sin \varphi \\
  x_n & = \pm r \sin \varphi \, \sqrt{1 - \sum_{k=1}^{n-2} \omega_k^2}.
\end{split}
\end{equation}
The expression for $x_n$ can be simplified to 
$x_n = r \omega_{n-1} \sin \varphi$ by considering
\begin{equation}
  \tilde{\omega} := (\omega_1, \omega_2, \ldots, \omega_{n-2}, \omega_{n-1})
  \in S^{n-2},
  \qquad
  \omega_{n-1} = \pm \sqrt{1 - \sum_{k=1}^{n-1} \omega_k^2}.
\end{equation}
When $r$ and $\varphi$ are fixed, 
$\tilde{\omega}$ runs over $S^{n-2}$ and $\omega$ runs twice over $B^{n-2}$,
so both cases of the $\pm$ sign must be covered. 
(See Figure~\ref{fig:spherical}.)

  \kuvienpoisto{
  \begin{figure}[htbp]
    \begin{center}
      \vspace{2ex}
      \leavevmode
      \epsfig{file=pics/spherical,width=12cm}
      \caption{Spherical coordinates in $\Rn$, $n \ge 3$.}
      \label{fig:spherical}
    \end{center}
  \end{figure}}


The Jacobian determinant is obtained as follows:
\begin{align}
%\begin{equation}
%\begin{split}
  \det J & 
  = \left| \begin{array}{cccccc}
      \dodo{x_1}{r} & \dodo{x_1}{\varphi} & \dodo{x_1}{\omega_1} &
        \dodo{x_1}{\omega_2} & \cdots & \dodo{x_1}{\omega_{n-2}} \\
      \dodo{x_2}{r} & \dodo{x_2}{\varphi} & \dodo{x_2}{\omega_1} &
        \dodo{x_2}{\omega_2} & \cdots & \dodo{x_2}{\omega_{n-2}} \\
      \dodo{x_3}{r} & \dodo{x_3}{\varphi} & \dodo{x_3}{\omega_1} &
        \dodo{x_3}{\omega_2} & \cdots & \dodo{x_3}{\omega_{n-2}} \\
      \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
      \dodo{x_{n-1}}{r} & \dodo{x_{n-1}}{\varphi} & \dodo{x_{n-1}}{\omega_1} &
        \dodo{x_{n-1}}{\omega_2} & \cdots & \dodo{x_{n-1}}{\omega_{n-2}} \\
      \dodo{x_n}{r} & \dodo{x_n}{\varphi} & \dodo{x_n}{\omega_1} &
        \dodo{x_n}{\omega_2} & \cdots & \dodo{x_n}{\omega_{n-2}}
    \end{array} \right| 
  \\ &
  \notag
  = \left| \begin{array}{cccccc}
    \cos\varphi & -r \sin\varphi & 0 & 0 & \cdots & 0 \\
      \omega_1\sin\varphi & r\omega_1\cos\varphi & r\sin\varphi & 0 & \cdots &
                                                                           0 \\
      \omega_2\sin\varphi & r\omega_2\cos\varphi & 0 & r\sin\varphi & \cdots &
                                                                           0 \\
      \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
      \omega_{n-2}\sin\varphi & r\omega_{n-2}\cos\varphi & 0 & 0 & \cdots &
                                                                r\sin\varphi \\
      \omega_{n-1}\sin\varphi & r\omega_{n-1}\cos\varphi & 
        -r \frac{\omega_1}{\omega_{n-1}} \sin\varphi & 
        -r \frac{\omega_2}{\omega_{n-1}} \sin\varphi & \cdots &
        -r \frac{\omega_{n-2}}{\omega_{n-1}} \sin\varphi 
  \end{array} \right|
%\end{split}
%\end{equation}
\end{align}
Subtracting the second column, multiplied by $\frac{\sin\varphi}{\cos\varphi}$,
from the first column and then developing the determinant with respect
to the first column, we get
%\begin{equation}
%\begin{split}
\begin{align}
  \det J & 
  = r^{n-1} \,\left| \begin{array}{cccccc}
      \frac{1}{\cos\varphi} & -\frac{\sin\varphi \cos\varphi}{\cos\varphi} & 
        0 & 0 & \cdots & 0 \\
      0 & \omega_1\cos\varphi & \sin\varphi & 0 & \cdots & 0 \\
      0 & \omega_2\cos\varphi & 0 & \sin\varphi & \cdots & 0 \\
      \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
      0 & \omega_{n-2}\cos\varphi & 0 & 0 & \cdots & \sin\varphi \\
      0 & \omega_{n-1}\cos\varphi & 
        - \frac{\omega_1}{\omega_{n-1}} \sin\varphi & 
        - \frac{\omega_2}{\omega_{n-1}} \sin\varphi & \cdots &
        - \frac{\omega_{n-2}}{\omega_{n-1}} \sin\varphi 
    \end{array} \right|
  \notag
  \\ &
  = r^{n-1} \sin^{n-2}\varphi \, \left| \begin{array}{ccccc}
      \omega_1 & 1 & 0 & \cdots & 0 \\
      \omega_2 & 0 & 1 & \cdots & 0 \\
      \vdots & \vdots & \vdots & \ddots & \vdots \\
      \omega_{n-2} & 0 & 0 & \cdots & 1 \\
      \omega_{n-1} & - \frac{\omega_1}{\omega_{n-1}} &
        - \frac{\omega_2}{\omega_{n-1}} & \cdots & 
        - \frac{\omega_{n-2}}{\omega_{n-1}}
  \end{array} \right|
\end{align}
%\end{split}
%\end{equation}
Now, subtracting the second through last column from the first,
multiplied by $\omega_1$, $\omega_2$, \ldots, $\omega_{n-2}$, respectively,
yields, since 
$\omega_1^2 + \omega_2^2 + \cdots + \omega_{n-2}^2 + \omega_{n-1}^2 = 1$,
\begin{equation}\nt
\begin{split}
  \det J & 
  = r^{n-1} \sin^{n-2}\varphi \, \left| \begin{array}{ccccc}
    0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\
    \vdots & \vdots & \vdots & \ddots & \vdots \\
    0 & 0 & 0 & \cdots & 1 \\ 
    \frac{1}{\omega_{n-1}} & -\frac{\omega_1}{\omega_{n-1}} &
    -\frac{\omega_2}{\omega_{n-1}} & \cdots& -\frac{\omega_{n-2}}{\omega_{n-1}}
    \end{array} \right|
  \\ &
  = (-1)^n \,
  \frac{r^{n-1} \sin^{n-2}\varphi}{\sqrt{1 - \sum_{k=1}^{n-2} \omega_k^2}}.
\end{split}
\ent\end{equation}
Thus,
\begin{equation}
  \label{SphericalCoordinatesRPhiOmega}
  \int_D f(x)\, dx = 
  \int_{D_r} \int_{D_\varphi} \int_{D_\omega} 
     \big(f_+(r,\varphi,\omega) + f_-(r,\varphi,\omega)\big)
     \, \frac{r^{n-1} \sin^{n-2}\varphi}{\sqrt{1 - \sum_{k=1}^{n-2} \omega_k^2}}
     \,d\omega \, d\varphi \, dr,
\end{equation}
where $f_+$ and $f_-$ correspond to the two signs in 
\equ{SphericalCoordinates}.
Analogously for $D \subset \Sm$, we have
\begin{equation}
  \label{SphericalCoordinatesPhiOmega}
  \int_D f(\theta)\, d\theta = 
  \int_{D_\varphi} \int_{D_\omega} 
     \big(f_+(\varphi,\omega) + f_-(\varphi,\omega)\big)
     \, \frac{\sin^{n-2}\varphi}{\sqrt{1 - \sum_{k=1}^{n-2} \omega_k^2}}
     \,d\omega \, d\varphi.
\end{equation}

In cases where the domain of integration $D_\omega$ is all of $B^{n-2}$
and $f$ does not depend on $\omega$ or the sign of $\omega_{n-1}$, 
the integral with respect to $\omega$ in 
\equ{SphericalCoordinatesRPhiOmega} and \equ{SphericalCoordinatesPhiOmega}
evaluates to $|S^{n-2}|$, as expected:

\begin{lemma} \label{lemma:IntSqrtOneMinusOmegaSquared}
\begin{equation}\nt
  2 \int_{B^{n-2}} \frac{d\omega}{\sqrt{1-|\omega|^2}}
  = |S^{n-2}|.
\ent\end{equation}
\end{lemma}
\begin{tod}
  Using the polar coordinates of $\R^{n-2} \supset B^{n-2}$,
  \begin{equation}\nt
    2 \int_{B^{n-2}} \frac{d\omega}{\sqrt{1-|\omega|^2}}
    = 2 \,|S^{n-3}|\, \int_0^1 \frac{r^{n-3}}{\sqrt{1-r^2}}\,dr.
  \ent\end{equation}
  On the other hand, 
  remembering Formula~\equ{MeasureUnitSphereAndBallLemma} for $|\Sm|$,
  we have
  \begin{equation}\nt
    2 \,|S^{n-3}|\,
      \frac{\sqrt{\pi} \,\Gamma\left(\frac{n-2}{2}\right)}{2
      \,\Gamma\left(\frac{n-1}{2}\right)}
    = 2 \, \frac{2 \pi^{(n-2)/2}}{\Gamma\left(\frac{n-2}{2}\right)} \,
      \frac{\sqrt{\pi} \,\Gamma\left(\frac{n-2}{2}\right)}{2
      \,\Gamma\left(\frac{n-1}{2}\right)}
    = \frac{2 \pi^{(n-1)/2}}{\Gamma\left(\frac{n-1}{2}\right)}      
    = |S^{n-2}|,
  \ent\end{equation}
  whence it suffices to show that 
  \begin{equation}\nt
    \label{lemma:IntSqrtOneMinusOmegaSquaredSufficesToShow}
    \int_0^1 \frac{r^{n-3}}{\sqrt{1-r^2}}\,dr 
    = \frac{\sqrt{\pi} \,\Gamma\left(\frac{n-2}{2}\right)}{2
            \,\Gamma\left(\frac{n-1}{2}\right)}.
  \ent\end{equation}
  To do this, we first note that since
  \begin{equation}\nt
  \begin{split}
    \int_0^1 \frac{r^k}{\sqrt{1-r^2}}\,dr &
    = \int_0^1 r^{k-1} \, \frac{r}{\sqrt{1-r^2}} \,dr
  \\ &
    = -\sij{0}{1} \left(r^{k-1} \sqrt{1-r^2}\right) 
    + (k-1) \int_0^1 r^{k-2} \sqrt{1-r^2}\,dr
  \\ &
    = (k-1) \int_0^1 \frac{r^{k-2}}{\sqrt{1-r^2}}\,(1-r^2) \,dr
  \\ &
    = (k-1) \int_0^1 \frac{r^{k-2}}{\sqrt{1-r^2}} \,dr
    - (k-1) \int_0^1 \frac{r^k}{\sqrt{1-r^2}} \,dr,
  \end{split}
  \ent\end{equation}
  we have the recursion formula
  \begin{equation}
    \label{IntRkDivBySqrtRecursion}
    \int_0^1 \frac{r^k}{\sqrt{1-r^2}} \,dr 
    =  \frac{k-1}{k}\, \int_0^1 \frac{r^{k-2}}{\sqrt{1-r^2}} \,dr
  \end{equation}
  for all $k \ge 2$. Now if $n$ is even,
  \begin{equation}\nt
    \begin{split}
    \Gamma\left(\frac{n-2}{2}\right) 
  & = \frac{n-4}{2}\,\Gamma\left(\frac{n-4}{2}\right) = \cdots
  \\ &
    = \frac{n-4}{2}\frac{n-6}{2} \cdots \frac{4}{2} \frac{2}{2} \,\Gamma(1)
    = \frac{n-4}{2}\frac{n-6}{2} \cdots \frac{4}{2} \frac{2}{2}
  \end{split}
  \ent\end{equation}
  and analogously
  \begin{equation}\nt
    \Gamma\left(\frac{n-1}{2}\right) = \frac{n-3}{2}\frac{n-5}{2} \cdots
    \frac{5}{2}\frac{3}{2}\frac{1}{2} \sqrt{\pi},
  \ent\end{equation}
  so \equ{IntRkDivBySqrtRecursion} gives us
  \begin{equation}\nt
  \begin{split}
    \int_0^1 \frac{r^{n-3}}{\sqrt{1-r^2}}\,dr 
  & = \frac{n-4}{n-3} \frac{n-6}{n-5} \cdots \frac{4}{5} \frac{2}{3} \,
      \int_0^1 \frac{r\,dr}{\sqrt{1-r^2}}
    \\ &
    = \frac{\frac{n-4}{2}\frac{n-6}{2} \cdots \frac{4}{2} \frac{2}{2}}{
        \frac{n-3}{2}\frac{n-5}{2} \cdots \frac{5}{2} \frac{3}{2}}
      \frac{\frac{\sqrt{\pi}}{2}}{\frac{1}{2}\sqrt{\pi}}
    = \frac{\sqrt{\pi} \,\Gamma\left(\frac{n-2}{2}\right)}{
        2 \,\Gamma\left(\frac{n-1}{2}\right)},
  \end{split}
  \ent\end{equation}
  as claimed. If $n$ is odd, we see similarly that
  \begin{equation}\nt
  \begin{split}
    \int_0^1 \frac{r^{n-3}}{\sqrt{1-r^2}}\,dr 
  & = \frac{n-4}{n-3} \frac{n-6}{n-5} \cdots \frac{3}{4} \frac{1}{2} \,
      \int_0^1 \frac{dr}{\sqrt{1-r^2}}
    \\ &
    = \frac{\frac{n-4}{2}\frac{n-6}{2} \cdots \frac{3}{2} \frac{1}{2}}{
        \frac{n-3}{2}\frac{n-5}{2} \cdots \frac{4}{2} \frac{2}{2}}
      \frac{\sqrt{\pi}}{2}
    = \frac{\sqrt{\pi} \,\Gamma\left(\frac{n-2}{2}\right)}{
        2 \,\Gamma\left(\frac{n-1}{2}\right)},
  \end{split}
  \ent\end{equation}
  which completes the proof.
\end{tod}

We have thus derived the following rule for evaluating integrals
using spherical coordinates:
\begin{lemma}
  \label{lemma:SphericalCoordinatesRn}
  If $n \ge 2$ and
  \begin{equation}
    D = \big\{(x_1, \ldots, x_{n-1}, \pm x_n) \,\big|\, 
              r \in D_r,\, \varphi \in D_\varphi,\, \omega \in D_\omega \big\} 
    \subset \Rn
  \end{equation}
  where
  \begin{equation}
  \begin{split}
  \label{SphericalCoordinatesLemma}
    x_1(r, \varphi, \omega) & = r \cos \varphi  \\
    x_2(r, \varphi, \omega) & = r \omega_1 \sin \varphi  \\
    x_3(r, \varphi, \omega) & = r \omega_2 \sin \varphi  \\
    & \hspace*{1.2ex} \vdots  \\
    x_{n-1}(r, \varphi, \omega) & = r \omega_{n-2} \sin \varphi  \\
    x_n(r, \varphi, \omega) & = 
      r \sin \varphi \, \sqrt{1 - \sum_{k=1}^{n-2} \omega_k^2},
  \end{split}
  \end{equation}
  $D_r \subset [0, \infty)$, $D_\varphi \subset [0, \pi]$,
  $D_\omega \subset B^{n-2}$, $f \in L^1(D)$ and
  \begin{equation}
    f_\pm(r, \varphi, \omega) 
    = f\big(x_1(r, \varphi, \omega), \ldots, x_{n-1}(r, \varphi, \omega),
            \pm x_n(r, \varphi, \omega)\big),
  \end{equation}
  then
  \begin{equation}
    \int_D f(x)\, dx 
    = 
    \int_{D_r} \int_{D_\varphi} \int_{D_\omega} 
    \big(f_+(r,\varphi,\omega) + f_-(r,\varphi,\omega)\big)
    \, \frac{r^{n-1} \sin^{n-2}\varphi}{\sqrt{1 - \sum_{k=1}^{n-2} \omega_k^2}}
    \,d\omega \, d\varphi \, dr.
  \end{equation}
  If furthermore $D_\omega = B^{n-2}$ and 
  $g(r, \varphi) = f_+(r, \varphi, \omega) = f_-(r, \varphi, \omega)$
  for all $r$, $\varphi$ and  $\omega$, then
  \begin{equation}
    \int_D f(x)\, dx
    = 
    |S^{n-2}| \int_{D_r} \int_{D_\varphi} g(r, \varphi)
      \, r^{n-1} \, \sin^{n-2}\varphi \,d\varphi \, dr.
  \end{equation}
\end{lemma}

Note that in the special case $n = 2$, we have $D_\omega = B^0 = \{0\}$, 
\begin{equation}
  \int_{D_\omega} \psi(r, \varphi, \omega) \,d\omega = \psi(r, \varphi, 0)
\end{equation}
using the counting measure, and
\begin{equation}
  \begin{cases}
    x = r \cos \varphi \\
    y = r \sin \varphi
  \end{cases}
\end{equation}
is just the familiar polar coordinate representation in the plane,
with the exception that here $\varphi \in [0, \pi]$ and
the two semicircles $S^1_+(\pm e_2)$ are covered using $f_\pm$.

When integrating over the unit sphere or part of it,
we have the following analogous result:
\begin{lemma}
  \label{lemma:SphericalCoordinatesSm}
  If $n \ge 2$ and
  \begin{equation}
    D = \big\{(\theta_1, \ldots, \theta_{n-1}, \pm\theta_n) \,\big|\,
              \varphi \in D_\varphi, \, \omega \in D_\omega \big\}
    \subset \Sm,
  \end{equation}
  where $\theta_i(\varphi, \omega) = x_i(1, \varphi, \omega)$ with $x_i$
  as in \equ{SphericalCoordinatesLemma} for all $i \in \{1, \ldots, n\}$,
  $f \in L^1(D)$ and
  \begin{equation}
    f_\pm(\varphi, \omega) 
    = f\big(\theta_1(\varphi, \omega), \ldots, \theta_{n-1}(\varphi, \omega),
        \pm \theta_n(\varphi, \omega)\big),
  \end{equation}
  then
  \begin{equation}
    \int_D f(\theta) \,d\theta
    = \int_{D_\varphi} \int_{D_\omega} 
        \big( f_+(\varphi, \omega) + f_-(\varphi, \omega) \big)
        \, \frac{\sin^{n-2}\varphi}{\sqrt{1 - \sum_{k=1}^{n-2} \omega_k^2}}
        \,d\omega \, d\varphi.
  \end{equation}
  If furthermore $D_\omega = B^{n-2}$ and 
  $g(\varphi) = f_+(\varphi, \omega) = f_-(\varphi, \omega)$
  for all $\varphi$ and  $\omega$, then
  \begin{equation}
    \int_D f(\theta)\, d\theta
    = 
    |S^{n-2}| \int_{D_\varphi} g(\varphi) \, \sin^{n-2}\varphi \,d\varphi.
  \end{equation}
\end{lemma}

The domains $D_r$, $D_\varphi$ and $D_\omega$ are here, 
for some sample domains $D$, the following:
\begin{center}
\begin{tabular}{|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|}
\hline 
D & D_r & D_\varphi & D_\omega \\
\hline 
\Rn & [0,\infty) & [0,\pi] & B^{n-2} \\
\hline 
\R^n_+ = \{ x\in\Rn \,|\, x_1 \ge 0\} & [0,\infty) & [0,\frac{\pi}{2}] 
                                                                & B^{n-2} \\
\hline 
\Sm & - & [0,\pi] & B^{n-2} \\
\hline 
S^{n-1}_+(e_1) \subset \Sm & - & [0,\frac{\pi}{2}] & B^{n-2} \\
\hline
\end{tabular}
\end{center}




\chapter{Table of Notation}
\label{app:notation}

\begin{longtable}{@{}>{$}l<{$}ll}
\textbf{Notation} & \textbf{Meaning} & \textbf{See page} \\[2ex]
\measuredangle & Angle & \tonref{def:measuredangle} \\
:=, =:    & Definition, assignment of value & \tonref{def:def} \\
*         & Convolution & \tonref{def:convolution}, 
                          \tonref{def:DistributionConvolution} \\
\sphat    & Fourier transform & \tonref{def:FourierTransform} \\
\sprefl   & Reflection & \tonref{def:reflection} \\
\theta^\bot
          & Orthogonal complement of $\theta$ 
                                        & \tonref{def:OrthogonalComplement} \\
\overline{A}
          & Closure of set A & \tonref{def:closure} \\
\conj{z}  & Complex conjugate of complex number $z$ \\
B^k       & Unit ball of $\R^k$ & \tonref{def:UnitBall} \\
B(x_0,R)  & Open ball with radius $R$ and centre at 
            $x_0$ & \tonref{def:BallRadiusRCentreX0} \\
C^k       & Space of $k$ times continuously differentiable functions &
                                                            \tonref{def:Ck} \\
C^k_0     & Space of $C^k$ functions with compact support &
                                                            \tonref{def:Ck} \\
D^\alpha,
D_j       & Distribution derivative & \tonref{def:DistributionDerivative} \\
\DD       & Divergent beam radiograph & \tonref{def:DivBeamRadiograph} \\
\DD       & Space of smooth functions with bounded support 
          & \tonref{def:TestFunctions} \\
\DD'      & Space of distributions & \tonref{def:Distributions} \\
\DD'_0    & Space of distributions with compact support 
          & \tonref{def:DistributionsWithCompactSupport} \\
\dist     & Distance & \tonref{def:dist} \\
\Dxr      & X-ray domain & \tonref{def:Dxr} \\
e_j       & $j^{\textup{th}}$ unit basis vector &\tonref{def:UnitBasisVector}\\
\mathcal{E}
          & Space of smooth functions
          & \tonref{def:DistributionsWithCompactSupport} \\
\mathcal{E'}
          & Space of distributions with compact support 
          & \tonref{def:DistributionsWithCompactSupport} \\
E_\theta  & Orthogonal projection onto $\theta^\bot$ 
          & \tonref{def:OrthogonalProjection} \\
\mathcal{F}
          & Fourier transform & \tonref{def:FourierTransform} \\
\mathcal{F}^{-1}
          & Inverse Fourier transform & \tonref{def:InverseFourierTransform}\\
H^s(\Rn)  & Sobolev space of order $s$ & \tonref{def:Hs} \\
K_\eps(\xi_0)
          & Cone with aperture $\eps$ and axis $\xi_0$
                                                &\tonref{def:cone} \\
L^p       & Space of functions whose $p^{\text{th}}$ power is integrable 
                                                   & \tonref{def:Lp} \\
L^p_{\textup{loc}} 
          & Space of functions whose $p^{\text{th}}$ power 
            is locally integrable                  & \tonref{def:Lploc} \\
L^\infty  & Space of essentially bounded functions & \tonref{def:Lp} \\
\N        & The set of natural numbers $\{0,1,2,\ldots\}$ & \tonref{def:N} \\
\PP       & Parallel beam radiograph & \tonref{def:ParBeamRadiograph} \\
\pv       & Principal value & \tonref{def:pvConvolution}, \tonref{def:pv} \\
\mathcal{R}
          & Radon transform & \tonref{def:RadonTransform} \\
R_1       & Riesz kernel & \tonref{def:RieszKernel} \\
\R        & Real field & \tonref{def:R} \\
\Rn       & $n$-dimensional space & \tonref{def:Rn} \\
\Ss       & Schwartz space of rapidly decreasing functions & \tonref{def:Ss}\\
\Ss'      & Space of tempered distributions & \tonref{def:Ssp} \\
\Sm       & Unit sphere in $\R^n$ & \tonref{def:UnitSphere} \\
\Sm(x_0,R)& Sphere with radius $R$ and centre $x_0$ & \tonref{def:sphere} \\
\spy      & Hemisphere closest to $y$ & \tonref{def:hemisphere} \\
\singsupp & Singular support & \tonref{def:singsupp} \\
\supp     & Support & \tonref{def:supp}\\
WF        & Wave front & \tonref{def:WaveFront} \\
\Z        & Set of integers $\{\ldots,-2,-1,0,1,2,\ldots\}$ &\tonref{def:Z} \\
\Z_+      & Set of positive integers $\{1,2,3,\ldots\}$ & \tonref{def:Zp} \\
\partial^\alpha,
\partial_j& Classical partial derivative & \tonref{def:PartialDerivative} \\
\partial A& Boundary of the set $A$ & \tonref{def:boundary} \\
\chi_X    & Characteristic function of the set $X$ & 
                         \tonref{def:CharacteristicFunction} \\
\tilde{\chi}_{\eps_1, \eps_2} &
            Approximate characteristic function of $B(0,\eps_1)$
          & \tonref{def:ApproxCharacteristicFunction} \\
\Lambda   & Calderón operator, inverse of $R_1 *$ & \tonref{def:Lambda} \\
\sigma_\lambda
          & Dilation & \tonref{def:dilation} \\
\end{longtable}

\newpage
\bibliographystyle{isa-kenny}
\label{bibliography}

\makeatletter
\addcontentsline{toc}{chapter}{\bibname}
\makeatother
\bibliography{dtyo}


\label{ViimeinenSivu}
\end{document}


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