TableOfFourierAndGeneralizedTransforms 2009-04-22 11:40:14 2009-04-22 11:49:58 Data Structure Fourier transform Fourier-Stieltjes transform Radon transform Laplace transform % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{tabls} % define commands here \usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym, enumerate} \usepackage{xypic, xspace} \usepackage[mathscr]{eucal} \usepackage[dvips]{graphicx} \usepackage[curve]{xy} \theoremstyle{plain} \newtheorem{lemma}{Lemma}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[section] \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem{example}{Example}[section] %\theoremstyle{remark} \newtheorem{remark}{Remark}[section] \newtheorem*{notation}{Notation} \newtheorem*{claim}{Claim} \renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote}}} 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\newcommand{\hr}{{\hookrightarrow}} \newcommand{\<}{{\langle}} %\newcommand{\>}{{\rangle}} \def\baselinestretch{1.1} \hyphenation{prod-ucts} %\grpeometry{textwidth= 16 cm, textheight=21 cm} \newcommand{\sqdiagram}[9]{$$ \diagram #1 \rto^{#2} \dto_{#4}& #3 \dto^{#5} \\ #6 \rto_{#7} & #8 \enddiagram \eqno{\mbox{#9}}$$ } \def\C{C^{\ast}} \newcommand{\labto}[1]{\stackrel{#1}{\longrightarrow}} %\newenvironment{proof}{\noindent {\bf Proof} }{ \hfill $\Box$ %{\mbox{}} \newcommand{\quadr}[4] {\begin{pmatrix} & #1& \\[-1.1ex] #2 & & #3\\[-1.1ex]& #4& \end{pmatrix}} \def\D{\mathsf{D}} \subsection{Table of Fourier and generalized transforms} \textbf{Fourier-Stieltjes} transforms and \textbf{measured groupoid} transforms are useful generalizations of the (much simpler) Fourier transform, as concisely shown in the following table. \subsubsection*{FT and FT Generalized Transforms} \begin{center} \begin{tabular}{|c|c|c|c|c|} \hline\hline $f(t)$ & $\F{f(t)} = \hat{f}(x)$ & Conditions* & Explanation & Description \\ \hline $c$ & $(\sqrt{2 \pi})^{-1}c$ & Notice on the next line the overline bar placed above $t(x)$ & &\\ & & \\ & & \\ \hline Gaussian function & Gaussian function & \\& &\\ & & \\ & & \\ \hline Lorentzian function & Lorentzian function & \\& &\\ & & \\ & & \\ \hline step function & $Sin(x)/x$ & \\& & \\ & & \\ & & \\ \hline sawtooth function & $Sin^2(x)/x^2$ & \\& & \\ & & \\ & & \\ \hline $f(t)$ & $\int \hat{f}(x) \overline{t(x)}dx$ & $f(t)\in{L^1(G_l)}$, with $G_l$ a & Fourier-Stieltjes transform & $\hat{f}(x)\in{C_0(\hat{G_l})}$ \\ & & locally compact groupoid \cite{RW97}; & & \\ & & $\int $ is defined \emph{via} & & \\ & & a left Haar measure on $G_l$ & & \\ \hline $\hat{m}(x)$ & $\check{m}(t)= \int e^{itx}d\hat{m}(x)$ & as above & Inverse Fourier-Stieltjes & $\check{m}(t) \in{L^1(G_l)}$, \\ & & & transform & (\cite{PALT2k1}, \cite{PALT2k3}). \\ \hline $\hat{m}(x)$ & $\check{m}(t) = \int e^{itx}d\hat{m}(x)$ & When $G_l=\mathbb{R}$, and it exists & This is the usual & $\check{m}(t) \in{\mathbb{R}}$ \\ & & only when $\hat{m}(x)$ is & Inverse Fourier transform & \\ & & \emph{Lebesgue integrable} on & & \\ & & the entire real axis & & \\ \hline\hline \end{tabular} \end{center} *Note the `slash hat' on $\hat{f}(x)$ and $\hat{G_l}$. \begin{thebibliography}{9} \bibitem{RW97} A. Ramsay and M. E. Walter, Fourier-Stieltjes algebras of locally compact groupoids, \emph{J. Functional Anal}. \textbf{148}: 314-367 (1997). \bibitem{PALT2k1} A. L. T. Paterson, The Fourier algebra for locally compact groupoids., Preprint, (2001). \bibitem{PALT2k3} A. L. T. Paterson, The Fourier-Stieltjes and Fourier algebras for locally compact groupoids., (2003) \PMlinkexternal{Free PDF file download}{http://aux.planetmath.org/files/objects/10739/AFourierStjelties_LocallyCompactsGds_Harmonic0310138v1.pdf}. \end{thebibliography}