%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: table of Fourier and generalized transforms
%%% Primary Category Code: 00.
%%% Filename: TableOfFourierAndGeneralizedTransforms.tex
%%% Version: 22
%%% Owner: bci1
%%% Author(s): bci1
%%% PlanetPhysics is released under the GNU Free Documentation License.
%%% You should have received a file called fdl.txt along with this file.
%%% If not, please write to gnu@gnu.org.
\documentclass[12pt]{article}
\pagestyle{empty}
\setlength{\paperwidth}{8.5in}
\setlength{\paperheight}{11in}
\setlength{\topmargin}{0.00in}
\setlength{\headsep}{0.00in}
\setlength{\headheight}{0.00in}
\setlength{\evensidemargin}{0.00in}
\setlength{\oddsidemargin}{0.00in}
\setlength{\textwidth}{6.5in}
\setlength{\textheight}{9.00in}
\setlength{\voffset}{0.00in}
\setlength{\hoffset}{0.00in}
\setlength{\marginparwidth}{0.00in}
\setlength{\marginparsep}{0.00in}
\setlength{\parindent}{0.00in}
\setlength{\parskip}{0.15in}
\usepackage{html}
% 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}}}
\numberwithin{equation}{section}
\newcommand{\Ad}{{\rm Ad}}
\newcommand{\Aut}{{\rm Aut}}
\newcommand{\Cl}{{\rm Cl}}
\newcommand{\Co}{{\rm Co}}
\newcommand{\DES}{{\rm DES}}
\newcommand{\Diff}{{\rm Diff}}
\newcommand{\Dom}{{\rm Dom}}
\newcommand{\Hol}{{\rm Hol}}
\newcommand{\Mon}{{\rm Mon}}
\newcommand{\Hom}{{\rm Hom}}
\newcommand{\Ker}{{\rm Ker}}
\newcommand{\Ind}{{\rm Ind}}
\newcommand{\IM}{{\rm Im}}
\newcommand{\Is}{{\rm Is}}
\newcommand{\ID}{{\rm id}}
\newcommand{\grpL}{{\rm GL}}
\newcommand{\Iso}{{\rm Iso}}
\newcommand{\rO}{{\rm O}}
\newcommand{\Sem}{{\rm Sem}}
\newcommand{\SL}{{\rm Sl}}
\newcommand{\St}{{\rm St}}
\newcommand{\Sym}{{\rm Sym}}
\newcommand{\Symb}{{\rm Symb}}
\newcommand{\SU}{{\rm SU}}
\newcommand{\Tor}{{\rm Tor}}
\newcommand{\U}{{\rm U}}
\newcommand{\A}{\mathcal A}
\newcommand{\Ce}{\mathcal C}
\newcommand{\D}{\mathcal D}
\newcommand{\E}{\mathcal E}
\newcommand{\F}{\mathcal F}
%\newcommand{\grp}{\mathcal G}
\renewcommand{\H}{\mathcal H}
\renewcommand{\cL}{\mathcal L}
\newcommand{\Q}{\mathcal Q}
\newcommand{\R}{\mathcal R}
\newcommand{\cS}{\mathcal S}
\newcommand{\cU}{\mathcal U}
\newcommand{\W}{\mathcal W}
\newcommand{\bA}{\mathbb{A}}
\newcommand{\bB}{\mathbb{B}}
\newcommand{\bC}{\mathbb{C}}
\newcommand{\bD}{\mathbb{D}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\bF}{\mathbb{F}}
\newcommand{\bG}{\mathbb{G}}
\newcommand{\bK}{\mathbb{K}}
\newcommand{\bM}{\mathbb{M}}
\newcommand{\bN}{\mathbb{N}}
\newcommand{\bO}{\mathbb{O}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bV}{\mathbb{V}}
\newcommand{\bZ}{\mathbb{Z}}
\newcommand{\bfE}{\mathbf{E}}
\newcommand{\bfX}{\mathbf{X}}
\newcommand{\bfY}{\mathbf{Y}}
\newcommand{\bfZ}{\mathbf{Z}}
\renewcommand{\O}{\Omega}
\renewcommand{\o}{\omega}
\newcommand{\vp}{\varphi}
\newcommand{\vep}{\varepsilon}
\newcommand{\diag}{{\rm diag}}
\newcommand{\grp}{\mathcal G}
\newcommand{\dgrp}{{\mathsf{D}}}
\newcommand{\desp}{{\mathsf{D}^{\rm{es}}}}
\newcommand{\grpeod}{{\rm Geod}}
%\newcommand{\grpeod}{{\rm geod}}
\newcommand{\hgr}{{\mathsf{H}}}
\newcommand{\mgr}{{\mathsf{M}}}
\newcommand{\ob}{{\rm Ob}}
\newcommand{\obg}{{\rm Ob(\mathsf{G)}}}
\newcommand{\obgp}{{\rm Ob(\mathsf{G}')}}
\newcommand{\obh}{{\rm Ob(\mathsf{H})}}
\newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}}
\newcommand{\grphomotop}{{\rho_2^{\square}}}
\newcommand{\grpcalp}{{\mathsf{G}(\mathcal P)}}
\newcommand{\rf}{{R_{\mathcal F}}}
\newcommand{\grplob}{{\rm glob}}
\newcommand{\loc}{{\rm loc}}
\newcommand{\TOP}{{\rm TOP}}
\newcommand{\wti}{\widetilde}
\newcommand{\what}{\widehat}
\renewcommand{\a}{\alpha}
\newcommand{\be}{\beta}
\newcommand{\grpa}{\grpamma}
%\newcommand{\grpa}{\grpamma}
\newcommand{\de}{\delta}
\newcommand{\del}{\partial}
\newcommand{\ka}{\kappa}
\newcommand{\si}{\sigma}
\newcommand{\ta}{\tau}
\newcommand{\lra}{{\longrightarrow}}
\newcommand{\ra}{{\rightarrow}}
\newcommand{\rat}{{\rightarrowtail}}
\newcommand{\ovset}[1]{\overset {#1}{\ra}}
\newcommand{\ovsetl}[1]{\overset {#1}{\lra}}
\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}}
\begin{document}
\subsection{Table of Fourier and generalized Fourier transforms}
\htmladdnormallink{Fourier transforms}{http://planetphysics.us/encyclopedia/FourierTransforms.html} are being very widely employed in physical, chemical and engineering applications for harmonic analysis, as well as for: processing acquired data such as spectroscopic, image processing (as for example in Astrophysics, elctron microscopy, optics), structure determination (e.g., \htmladdnormallink{X-ray}{http://planetphysics.us/encyclopedia/FluorescenceCrossCorrelationSpectroscopy.html}, \htmladdnormallink{neutron}{http://planetphysics.us/encyclopedia/Pions.html}, electron diffraction), chemical \htmladdnormallink{Hyperspectral Imaging}{http://planetphysics.us/encyclopedia/SpectralImaging.html} (FT-NIR, FT-IR), and so on. Theoretical studies in \htmladdnormallink{quantum mechanics}{http://planetphysics.us/encyclopedia/QuantumParadox.html} ({\em QM}), \htmladdnormallink{QCD}{http://planetphysics.us/encyclopedia/LQG2.html}, \htmladdnormallink{QG}{http://planetphysics.us/encyclopedia/LQG2.html}, \htmladdnormallink{AQFT}{http://planetphysics.us/encyclopedia/MetaTheorems.html}, \htmladdnormallink{quantum theories}{http://planetphysics.us/encyclopedia/SpaceTimeQuantizationInQuantumGravityTheories.html} on a lattice ({\em QTL}) also employ Fourier transforms.
\textbf{Fourier-Stieltjes} transforms and \textbf{measured \htmladdnormallink{groupoid}{http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html}} transforms are useful generalizations of the (much simpler) Fourier transform, as concisely shown in the following table.
\subsubsection*{Fourier Transforms and Generalized FTs}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline\hline
$f(t)$ & $\F{f(t)} = \hat{f}(x)$ & Conditions* & Explanation & Description \\
\hline
Gaussian \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} & Gaussian function & general & In statistics,& and also in spectroscopy \\
\hline
\htmladdnormallink{Lorentzian}{http://planetphysics.us/encyclopedia/LebesgueMeasure.html} function & Lorentzian function & general & In spectroscopy & experimentally truncated to the single exponential function with a negative exponent \\
\hline
step function & $sin(x)/x$ & general & FT of a \htmladdnormallink{square}{http://planetphysics.us/encyclopedia/PiecewiseLinear.html} \htmladdnormallink{wave}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} & `slit' function \\
\hline
sawtooth function & $sin^2(x)/x^2$ & general& a triangle & zero baseline \\
\hline
series of equidistant points ....& (inf.) \htmladdnormallink{group}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of equidistant planes & general & lattice of infinite planes & used in diffraction theory\\
\hline
lattice of infinite planes, (or 1D paracrystal) & series of equidistant points .... & general & one-dimensional reciprocal space & used in crystallography/diffraction theory\\
\hline
Helix wrapped on a cylinder & \htmladdnormallink{Bessel functions/}{http://planetphysics.us/encyclopedia/BesselEquation2.html} series& general & In Physical Crystallography & experimentally truncated to the first (finite)
n-th order Bessel functions\\
\hline
$c$ & $(\sqrt{2 \pi})^{-1}c$ & Notice on the next line the overline bar placed above $t(x)$ & general & Integration constant\\
\hline
$f(t)$ & $\int \hat{f}(x) \overline{t(x)}dx$ & $f(t)\in{L^1(G_l)}$, with $G_l$ a & \htmladdnormallink{Fourier-Stieltjes transform}{http://planetphysics.us/encyclopedia/StieltjesTransform.html} & $\hat{f}(x)\in{C_0(\hat{G_l})}$ \\
& & \htmladdnormallink{locally compact groupoid}{http://planetphysics.us/encyclopedia/LocallyCompactGroupoid.html} \cite{RW97}; & & \\
& & $\int $ is defined \emph{via} & & \\
& & a left \htmladdnormallink{Haar measure}{http://planetphysics.us/encyclopedia/HigherDimensionalQuantumAlgebroid.html} 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) \htmladdnormallink{Free PDF file download}{http://aux.planetmath.org/files/objects/10739/AFourierStjelties_LocallyCompactsGds_Harmonic0310138v1.pdf}.
\end{thebibliography}
\end{document}
