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\title{Integral Transforms}
\author{Roe Goodman}                 % Enter your name between curly braces
\institute{Introduction to Math at Rutgers }      % Enter your institute name between curly braces
\date{August 29, 2010}             % Enter the date or \today between curly braces


%%%%%%%%%%%%%  end of preamble

\begin{document}

\begin{frame}
  \titlepage
\end{frame}
\note{ } % Add notes to yourself that will be displayed when
                           % typeset with the notes class option

% \section{Outline}

% Creates table of contents slide incorporating
% all \section and \subsection commands
% \begin{frame}
%  \tableofcontents[pausesections]

% \pause


%\end{frame}

%%%%%%%%%%%% frame 1 %%%%%%%%%%

\begin{frame}
\frametitle{Fourier Transform}

$G $ -- locally compact abelian topological group (written additively)

\pause

$G$ has a \alert{Haar measure} (translation invariant)

$L^2(G)$ -- square integrable complex valued functions on $G$ 

\quad Inner product: \quad
%
$(\phi, \psi) = \int_G \phi(x) \overline{\psi(x)} \, \md x$   

\quad norm: \quad $\|\phi\|_2 = \sqrt{(\phi, \phi)}$


$G$ acts on $L^2(G)$ by translations: \ $T_y\phi(x) = \phi(x-y)$

\pause

\begin{definition} A linear transformation (operator) $C: L^2(G) \to L^2(G)$
is \alert{translation invariant} if it commutes with $\{T_y\}_{y \in G}$.
 \end{definition}

\pause
 
Some Examples

\begin{itemize}

\item
\alert{Translation:} $C\phi = T_y \phi$ with  $y \in G$
 
\item
\alert{Convolution:} 
%
 $ C\phi(x) = \int_G f(y)\phi(x-y) \, dy$ with  $f \in L^1(G)$

(weighted average of translates of $\phi$) 

\end{itemize}

\pause

\alert{Problem: } Diagonalize all translation invariant operators 

\pause

\alert{Solution:} Use \alert{characters} of $G$ and \alert{Fourier transform}

\end{frame}


%%%%%%%%%%%   frame 2 %%%%%%%%%%%%%%%
\begin{frame}
  \frametitle{Fourier Transform on $\bZ/n\bZ$}

\alert{Example 1} \quad $G = \bZ/n\bZ$ (additive group of integers mod $n$)

$L^2(G) =
 \{ \phi: \bZ \to \bC \;:\; \phi(k+n) = \phi(k) \mbox{\ for all $k \in \bZ$}\}$

\vspace{1ex}

inner product $(\phi, \psi) 
  = \frac{1}{n}\sum_{k = 0}^{n-1} \phi(k)\overline{\psi(k)}$         

\pause

\vspace{1ex}

\alert{Characters} \quad $e_p(k) = w^{kp}$ for $k, p \in \bZ$ \quad ($w =
\me^{2\pi \mi/n}$, \ $w^n = 1$)

\pause

\begin{itemize}

\item
 $e_p(k+m) = e_p(k)e_p(m)$, \quad  $|e_p(k)| = 1$, \quad $e_{p+n} = e_p$

\item
Eigenfunctions for translations \quad $T_ke_p = w^{-kp}e_p$

\item
Orthogonality relations \quad 
$ (e_p, e_q)
 = \left\{\begin{array}{ll} 
    1 & \mbox{if $p - q \equiv 0 \mod(n)$}
    \\
    0 & \mbox{else}
    \end{array} \right.$

\end{itemize}

\pause

\alert{Finite Fourier Transform} 
\quad $\widehat{\phi}(p) = (\phi, e_p)$ 

\pause

\begin{itemize}

\item
Diagonalization 
\quad
$\psi = T_k\phi$  \quad$\Rightarrow$\quad 
  $\widehat{\psi}(p) = w^{-kp}\widehat{\phi}(p)$

\item
Fourier inversion  
\quad $\phi =  \sum_{p=0}^{n-1} \widehat{\phi}(p) e_p$

\item
Plancherel formula  
\quad $(\phi, \psi) 
 =  \sum_{p=0}^{n-1} \widehat{\phi}(p)\overline{\widehat{\psi}(p)}$

\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%% frame 3 %%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Diagonalization of Translation Invariant Operators}

\begin{theorem}
Let $G = \bZ/n\bZ$. Let $C$ be a translation invariant operator on $L^2(G)$.
There is a function $F$ on $\widehat{G} \cong \bZ/n\bZ$ such that 

$(\star)$\qquad $ \widehat{C\phi}(p) = F(p)\widehat{\phi}(p)$ 
for all $\phi \in L^2(G)$ and $p \in \bZ$.

\noindent 
Conversely, every function $F$ on $\bZ/n\bZ$ defines a translation invariant
operator $C$ on $L^2(G)$ by $(\star)$ ($C =$ convolution by $f$, where
$\widehat{f} = F$).
\end{theorem}

\pause

\begin{proof}  
Let $S = T_1$ (shift operator). Then $S$ has $n$ distinct eigenvalues
$\lambda_p = w^{-p}$ for $p = 0, \ldots, n-1$ with eigenvectors $e_p$. Since
$C$ commutes with $S$, the function $Ce_p$ is an eigenvector for $S$ with
eigenvalue $w^{-p}$. Hence
%
$Ce_p = F(p)e_p$ for some scalar $F(p) \in \bC$.  
%
The Fourier inversion formula now implies $(\star)$.
\end{proof} 

\end{frame}
%%%%%%%%%%%%%%%%%%%% frame 4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
  \frametitle{Character Group and Duality }

\alert{General Version of Fourier Transform}

$G $ -- locally compact abelian topological group (written additively)

$\widehat{G}$ -- all \alert{characters} of $G$: 
   
\quad  $\psi: G \to \bT = \{z \in \bC \;:\; |z| = 1 \}$ (continuous)

\quad $\psi(x+y) = \psi(x)\psi(y)$, \quad $\psi(0) = 1$

\pause

\begin{itemize}

\item 
$\widehat{G}$ is a locally compact abelian group under pointwise
multiplication and uniform convergence on compacta topology.

\pause

\item
$\widehat{(\widehat{G})} \cong G$ \ (\alert{natural} isomorphism, as for vector
space duality).

\pause

\item
Fourier transform takes $L^2(G)$ onto $L^2(\widehat{G})$ preserving norm.

\pause

\item
Translation invariant operator $C$ on $L^2(G)$ becomes multiplication by a function $F$
on $L^2(\widehat{G})$. 

\end{itemize}

\pause

\alert{Example}

$G = \bZ/n\bZ$ 
% 
\quad $\widehat{G} = \{ e_p \;:\; p \in \bZ/n\bZ \} \cong G$ 

Choose \alert{basic} character $e_1$. Then  $e_p(k) = e_1(pk)$

\end{frame}
%%%%%%%%%%%%%%%%%%%% frame 5 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
  \frametitle{Fourier Transform on $\bR/\bZ$ }

\alert{Example 2} \quad  
$G = \bR/\bZ$ (additive group of real numbers modulo $1$)

$L^2(\bR/\bZ) \quad 
 \phi: \bR \to \bC, \  \phi(x+1) = \phi(x)$
 \ (periodic, measurable)

\qquad \qquad  $\int_0^1 |\phi(x)|^2 \, \md x < \infty$ \ (Lebesgue integral)

Inner product 
\quad $(\phi, \psi) = \int_0^1 \phi(x)\overline{\psi(x)} \, \md x$         

\vspace{1ex}

\pause

\alert{Characters} \quad $e_p(x) = \exp(2\pi\mi px)$ for $p \in \bZ$
and $x \in \bR$

\pause

\begin{itemize}

\item
 $e_p(x+1) = e_p(x)$, \quad  $|e_p(x)| = 1$

\item
 $e_p(x)e_q(x) = e_{p+q}(x)$, so $\widehat{G} \cong \bZ$ under  $e_p
 \leftrightarrow p$

\item
Orthogonality relations \quad 
$ (e_p, e_q)
 = \left\{\begin{array}{ll} 
    1 & \mbox{if $p = q$}
    \\
    0 & \mbox{else}
    \end{array} \right.$

\end{itemize}

\pause

\alert{Fourier transform} 
\quad $\widehat{\phi}(p) = (\phi, e_p) 
  = \int_{0}^{1} \phi(x) \exp(-2\pi\mi px) \, \md x$

\pause

\begin{itemize}

\item
Diagonalization 
\quad
$\psi = T_y\phi$  \quad$\Rightarrow$\quad 
  $\widehat{\psi}(p) =  e_p(-y) \widehat{\phi}(p)$

\item
Fourier inversion  
\quad $\phi = \sum_{p \in \bZ} \widehat{\phi}(p) e_p$
%
\quad($L^2$ convergence)

\item
Plancherel formula  
\quad $(\phi, \psi) 
 = \sum_{p \in \bZ} \widehat{\phi}(p)\overline{\widehat{\psi}(p)}$

\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%% frame 6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Bounded Translation Invariant Operators}

Linear operator $C$ on $L^2(\bR/\bZ)$ is \alert{bounded} if 
  $\|C\phi\|_2 \leq M \|\phi\|_2$ 

Same as: $C$ is a \alert{continuous} transformation w.r.t. $\|\phi\|_2$.

\pause

\begin{theorem}
Let $C$ be a \alert{bounded} translation invariant operator on $L^2(\bR/\bZ)$. Then there
is a \alert{bounded} function $F$ on $\bZ$ such that 

$(\star)$ \qquad 
$ \widehat{C\phi}(p) = F(p)\widehat{\phi}(p)$ for all $\phi \in L^2(\bR/\bZ)$.

\vspace{.5ex}

\noindent 
Conversely, every bounded function $F$ on $\bZ$ defines a bounded
translation invariant operator $C$ on $L^2(\bR/\bZ)$ by $(\star)$.
\end{theorem}

\pause

\begin{proof}
Let $S = T_y$, $y$  \alert{irrational}. Then $S$ has \alert{distinct}
eigenvalues $\lambda_p = \exp(-2\pi\mi yp)$ for $p \in \bZ$ with eigenvectors
$e_p$. 

\pause

$CS = SC$ $\Rightarrow$ 
%
$Ce_p = F(p)e_p$ with  $F(p) \in \bC$.  
%
Then $C$  bounded \ $\Rightarrow$ 
 \ $\|F\|_{\infty} := \sup_p |F(p)| < \infty$.
%
Hence

\qquad $C\phi = \sum_{p \in \bZ} \widehat{\phi}(p) Ce_p =
   \sum_{p \in \bZ} \widehat{\phi}(p)F(p)e_p $
\end{proof}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%% frame 7 %%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Fourier Analysis of $C^{\infty}(\bR/\bZ)$}

$C^{\infty}(\bR/\bZ) =$ \ differentiable periodic functions on $\bR$

\pause

$D = \frac{1}{2\pi\mi}\frac{d}{dx}$  \ 
translation invariant operator on $C^{\infty}(\bR/\bZ)$


\begin{itemize}

\item   
$De_p = p e_p$ \ for $p \in \bZ$, so $D$ is \alert{not bounded} on $L^2(\bR/\bZ)$ 

\pause

\item   
 $(D\phi, \psi) = (\phi, D\psi)$ for $\phi, \psi \in C^{\infty}(\bR/\bZ)$
(integrate by parts)

\item
$\widehat{D\phi}(p) = p\widehat{\phi}(p)$ for $\phi \in C^{\infty}(\bR/\bZ)$

\item
$\phi \in C^{\infty}(\bR/\bZ) 
\Longleftrightarrow \widehat{\phi}$ is \alert{rapidly
 decreasing}: 

\vspace{.5ex}
 
For every positive integer $r$ \quad
 $\sup_{p\in \bZ} |p^r \widehat{\phi}(p)| < \infty$
  
\end{itemize}

\pause

\begin{theorem}
Let $C$ be a \alert{continuous} translation invariant operator on
$C^{\infty}(\bR/\bZ)$. Then there is a function $F$ on $\bZ$ of
\alert{polynomial growth} at $\infty$ such that 

$(\star)$ \qquad 
$ \widehat{C\phi}(p) = F(p)\widehat{\phi}(p)$ 
for all $\phi \in C^{\infty}(\bR/\bZ)$.

\noindent 
Conversely, every such function $F$ on $\bZ$ defines a continuous translation
invariant operator $C$ on $C^{\infty}(\bR/\bZ)$ by $(\star)$.
\end{theorem}

\end{frame}

%%%%%%%%%%%%%%%%%%%% frame 8 %%%%%%%%%%%%%%%%%%%
\begin{frame}
  \frametitle{Fourier Transform on $\bR$}

\alert{Example 3} \quad  
$G = \bR$ (additive group of real numbers)

$L^2(\bR) \quad  \phi: \bR \to \bC$,  \ (measurable) \ 
%
$\int_{-\infty}^{\infty} |\phi(x)|^2 \, \md x < \infty$ 

Inner product 
\quad $(\phi, \psi) = 
   \int_{-\infty}^{\infty} \phi(x)\overline{\psi(x)} \, \md x$         

\vspace{1ex}

\pause

\alert{Characters} \quad $e_{\xi}(x) = \exp(2\pi\mi x \xi )$ 
  for $x, \xi \in \bR$.

\pause

\begin{itemize}

\item
Fix \alert{basic} character $e_1$. Then $e_{\xi}(x) = e_1(x\xi)$
 
\item 
$e_{\xi}(x)e_{\tau}(x) = e_{\xi + \tau}(x)$, so $\widehat{\bR} \cong \bR$
under $e_{\xi} \leftrightarrow \xi$

\item
$\bR$ \alert{ not compact }  $\Rightarrow e_{\xi} \notin L^2(\bR)$  
(plane wave, frequency $\xi$)

\end{itemize}

\pause

\alert{Fourier transform} For $\phi \in L^1(\bR)$ define

\vspace{.5ex}

\quad $\widehat{\phi}(\xi) = 
   \int_{-\infty}^{\infty} \phi(x) e_{-\xi}(x) \, \md x$
%
\ (integral converges absolutely)

\pause

\begin{itemize}

\item
Fourier transform extends to isometry  $L^2(\bR) \to L^2(\widehat{\bR})$

\item
Plancherel formula  
\quad $(\phi, \psi) 
 = \int_{-\infty}^{\infty} 
  \widehat{\phi}(\xi)\overline{\widehat{\psi}(\xi)}\, \md\xi$

\item 
Bounded translation invariant operator $C$ on $L^2(\bR)$ $\longleftrightarrow$
multiplication by bounded measurable function $F$ on $\widehat{\bR}$

\end{itemize}

\end{frame}


%%%%%%%%%%%%%%%%%%%% frame 9 %%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Tempered Fourier Analysis on $\bR$}

$\cS(\bR) =$ \ \alert{rapidly decreasing}
 $C^{\infty}$ functions on $\bR$:

\qquad
$\sup_{x\in \bR} \left|x^m \left(\frac{d}{dx}\right)^k\phi(x)\right| < \infty$ 
for all positive integers $m, k$


\alert{Example}  \quad $\phi(x) = p(x)e^{-\pi x^2}$ \ with $p(x)$ a polynomial

Fourier transform of $\phi$ is $q(\xi)e^{-\pi \xi^2}$  
with $q$ a polynomial

\pause

\begin{itemize}

\item   
$\cS(\bR)$ invariant under $D_x = \frac{1}{2\pi\mi}\frac{d}{dx}$, \ $M_x =$
multiplication by $x$

\item   $\widehat{D_x\phi} = M_{\xi}\widehat{\phi}$ 
 for $\phi \in \cS(G)$ (integrate by parts)

\item   $\widehat{M_x\phi} = D_{\xi}\widehat{\phi}$ 
 for $\phi \in \cS(G)$ (differentiate under integral)

\item
 $\phi \in \cS(\bR) \Longleftrightarrow \widehat{\phi} \in \cS(\bR)$
  
\end{itemize}

\pause

\begin{theorem}
Let $C$ be a \alert{continuous} translation invariant operator
on $\cS(\bR)$. Then there is a $C^{\infty}$ function $F$ on $\bR$ with all derivatives of
\alert{polynomial growth} at $\infty$ such that 

$(\star)$ \qquad 
$ \widehat{C\phi}(\xi) = F(\xi)\widehat{\phi}(\xi)$ 
for all $\phi \in \cS(\bR)$.

\noindent 
Conversely, every such function $F$ on $\bR$ defines a
continuous translation invariant operator $C$ on $\cS(\bR)$ by $(\star)$.
\end{theorem}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%  frame 10 %%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Fourier-Mellin Transform}

$\bR^{\times} = \bR\setminus \{0\}$ 
-- locally compact group under multiplication

\begin{itemize}

\item
Invariant integral \quad $
\displaystyle{ \int_{-\infty}^{\infty} f(x) \, \frac{\md x}{|x|} }$

\item
Characters \quad 
$e_{\tau, \epsilon}(x) =  \sgn(x)^{\epsilon} |x|^{\mi \tau}$ 
\ with  $\tau \in \bR$ and  $\epsilon = \pm 1$

$\widehat{\bR^{\times}} \cong \bR \times (\bZ/2\bZ)$

\pause

\item
Fourier-Mellin transform
\quad $\widehat{f}(\tau, \epsilon) 
 = \displaystyle{\int_{-\infty}^{\infty} f(x) e_{-\tau, \epsilon}(x) \, 
  \frac{\md x}{|x|} }$

for $f \in L^1(\bR, \frac{\md x}{|x|})$

\item
Plancherel Formula

\vspace{1ex}

\qquad $
\displaystyle{\int_{-\infty}^{\infty} f(x) \overline{g(x)} \,  \frac{\md x}{|x|}
 = \sum_{\epsilon = \pm 1}\int_{-\infty}^{\infty} 
\widehat{f}(\tau,\epsilon)\overline{\widehat{g}(\tau,\epsilon)} \, \md\tau }$

for $f, g  \in L^2(\bR, \frac{\md x}{|x|})$

\end{itemize}

\pause

\alert{Log Trick:} Use group homomorphism $x \mapsto (\log |x|, \sgn(x))$ to
turn Fourier-Mellin transform into Fourier transform on $\bR \times (\bZ/2\bZ)$.

\end{frame}


%%%%%%%%%%%%%%%%%%%%%%% frame 11 %%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Fourier Analysis on $\bQ_p$}

$\bQ_p = $ completion of $\bQ$ relative to $p$-adic absolute value ($p$ prime)

 \quad $|p^k r/s|_p = p^{-k}$  if $r$, $s$ integers relatively prime to $p$

\pause

\begin{itemize}

\item  locally compact \alert{totally disconnected} field with
 metric $d(x,y) = |x - y|_p$,  \qquad $|x + y|_p = \max\{|x|_p, |y|_p\}$

\pause

\item
\alert{$p$-adic expansion} \quad  $x = \sum_{n = k}^{\infty} a_n p^n$  
\quad
 $a_n \in \{0, 1, \ldots, p-1 \}$  

$|x|_p = p^{-k}$ with $k = \min \{ n \;:\; a_n \neq 0 \}$ if $x \neq 0$

\pause

\item
ring of \alert{$p$-adic integers} \quad $\bZ_p = \{ |x|_p \leq 1 \}$ (compact)

\end{itemize}

\pause

\alert{Characters} Let $\bQ_p^{+} =$   additive group of $\bQ_p$ 

\begin{itemize}

\item 
$e(x) =  \exp(2\pi\mi z)$ 
with $z = \sum_{n < 0} a_n p^n \in \bQ$ 
%
\quad ($x \in z + \bZ_p$)  

\pause

\item
$\bQ_p^{+} \cong \widehat{\bQ_p^{+}} 
= \{e_y \}_{y \in \bQ_p}$ where  $e_y(x) = e(xy)$ 
for $x, y \in \bQ_p$ 

% (note that $xy \in \bQ \mod \bZ_p$ and $e(\bZ_p) = 1$)

\pause

\item
Fourier transform analogous to Fourier transform on $\bR^{+} =
\bQ_{\infty}^{+}$

\end{itemize}

\pause

\alert{Fourier-Mellin} transform on $\bQ_p^{\times}$  
more complicated than $\bR^{\times}$

\pause

\quad $\bQ_p^{\times} \cong \{p^k\}_{k\in \bZ} \times (\bZ/(p-1)\bZ)\times A$
 with $A = \exp\{x \;:\;  |x|_p < 1 \}$ 

\pause
 
\quad  
   $\widehat{\{p^k\}_{k\in \bZ}} \cong \bR/\bZ$ \ (compact) 
\quad  
 $\widehat{A} \cong \displaystyle{\lim_{k \to \infty}^{\longleftarrow}} \; 
\bZ/(p^k\bZ)$ \ (countable) 

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%% frame 12 %%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Laplace operator} 

$G =$ Euclidean motion group on $\bR^n$ (translations and rotations)

\vspace{.5ex}

\pause

$\Delta = \partial^2/ \partial x_1^2 + \cdots + \partial^2/ \partial x_n^2$  
 \quad \alert{Laplace operator} on $\bR^n$

\vspace{.5ex}


Polynomials in $\Delta$ give all differential operators on $\bR^n$
invariant under $G$

\vspace{1ex}

\pause

\alert{Problem}: Diagonalize action of $\Delta$ on $L^2(\bR^n)$

\vspace{1ex}

\pause

\alert{Fourier Transform Method:} 

Use spherical coordinates on $\bR^n$ (singularity at $0$) and expansion in
spherical harmonics. On radial functions get \alert{Fourier-Bessel}
transform (integral transform with Bessel function kernel).

\vspace{1ex}

\pause

\alert{Radon Transform Method:}  

Use integral transform that turns $\Delta$ into $(\partial/\partial p)^2$ on
\alert{even} functions of $p \in \bR$ with \alert{parameter} $\omega \in \bS^{n-1}$ (no
singularity). Then diagonalize by \alert{one-dimensional} Fourier transform.

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%  frame 13 %%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Radon Transform}


$\bS^{n-1} = \{ x \in \bR^n \;:\; x\cdot x = 1 \}$  unit sphere

$x\cdot y = x_1y_1 + \cdots + x_ny_n$ \quad inner product on $\bR^n$

\pause

Hyperplane with \alert{oriented normal} $\omega \in \bS^{n-1}$ and
\alert{height} $p \in \bR$: 

\quad $H(\omega, p) = \{ x \in \bR^n \;:\;  x\cdot \omega = p \}$

\pause

Write $\xi = H(\omega, p) \cong \bR^{n-1}$

\quad $\md m = (n-1)$-dimensional Lebesgue measure on $\xi$

\pause


$\bP^n = $ \ set of all hyperplanes $\xi$ in $\bR^n$ (smooth $n$-dim manifold)

\alert{two-sheeted covering} $\bS^{n-1}\times \bR \to \bP^n$  (no singularities)

\quad$(\omega, p) \mapsto H(\omega, p)  =  H(-\omega, -p)$  

\pause

\alert{Radon transform} of $f \in \cS(\bR^n)$: 

\vspace{.5ex}

\quad $F(\omega, p) = \int_{x\cdot\omega = p}  f(x) \, \md m(x) $

\pause

\begin{itemize}

\item
Integral converges since $\left.f\right|_{H(\omega, p)}$ is rapidly
decreasing

\item
$F(\xi) = F(\omega, p) $ 
defined on $\bP^n$ since $F(\omega, \xi) = F(-\omega, -\xi)$

\pause

\item
Fourier transform $\widehat{f}(r\omega) 
= \int_{-\infty}^{\infty} F(\omega, p) \me^{-2\pi\mi rp} \; \md p$

\pause

\item
Radon transform of $\Delta f(x)$ is 
$\left(\partial/\partial p\right)^2 F(\omega, p) $

\end{itemize}

\end{frame}


%%%%%%%%%%%%%%%%%%% frame 14 %%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Inverse Radon Transform }

For $x\in \bR^n$

\ \  $K(x) = $ all hyperplanes $\xi$ containing $x$ 

\quad\qquad $= \{ (\omega, p) \;:\; x\cdot \omega = p \} 
 \cong \bS^{n-1}/\pm 1$

Let $\md\mu = $ invariant measure on $K(x)$ (total mass $1$)

\pause

For $F \in \cS(\bP^n)$ define \alert{dual Radon transform}  

\vspace{1ex}

\quad $\widetilde{F}(x) 
= \int_{\xi \in K(x)} F(\xi)\, \md\mu(\xi) 
= \int_{\omega \in \bS^{n-1}} F(\omega, \omega\cdot x)\, \md\omega $

\vspace{1ex}

\pause

\alert{Radon Inversion Formula}

If $f\in \cS(\bR^n)$ and  $F = $ Radon transform of $f$, then 

\vspace{.5ex}

\quad
 $ f(x) = c\left(-\Delta\right)^{(n-1)/2} \widetilde{F}(x)$ 
 \qquad ($c =$ normalizing constant)

\vspace{1ex}

\pause
 
\alert{odd dimensions:} Inversion formula is \alert{local} - differential operator applied to $\widetilde{F}(x)$

\alert{even dimensions:} Inversion formula is \alert{non-local} - square root
of differential operator (Hilbert transform) applied to $\widetilde{F}(x)$

\end{frame}

\begin{frame}
\frametitle{Further Reading}


The Wikipedia articles on Fourier Analysis, p-adic Numbers, and Radon
Transform are good starting points.  Here are some books:

\begin{itemize}

\item
Fourier analysis on locally compact abelian groups:

W. Rudin, {\bf Fourier Analysis on Groups}, Wiley (1962)

G. Folland, {\bf A Course in Abstract Harmonic Analysis}, CRC Press (1995)

\item
Finite Fourier transform:

A. Terras, {\bf Fourier Analysis on Finite Groups and applications}, Cambridge (1999)

\item
Fourier analysis on $\bR/\bZ$ and $\bR$:

G. Folland, {\bf  Real Analysis: Modern
Techniques and Their Applications}, Wiley (1999)


\item
Radon Transform:

S. Helgason, {\bf Groups and Geometric Analysis}, Academic Press (1984)

\end{itemize}

\end{frame}


\end{document}


