table of Fourier and generalized transforms TableOfFourierAndGeneralizedTransforms 2009-04-22 11:40:14 2026-09-09 21:46:29 Data Structure section numbering quantum theories on a lattice QM QTL FT-NIR FT-IR Fourier transform Fourier-Stieltjes transform table of Fourier and generalized transforms Radon transform Laplace transform FT-NIR FT-IR QCD QG QFT QLT AQFT quantum theories on a lattice % Object 685: Table of Fourier and Generalized Transforms % Minimal pdflatex/make4ht-safe preamble. % % Removed obsolete/missing tabls.sty and the large unused legacy package/macro stack. \usepackage{amsmath} \usepackage{amssymb} \usepackage{array} \subsection{Table of Fourier and Generalized Fourier Transforms} Fourier transforms are widely employed in physical, chemical, and engineering applications for harmonic analysis and for processing acquired data, such as spectroscopic data and images. Applications include astrophysics, electron microscopy, optics, structure determination (for example, X-ray, neutron, and electron diffraction), chemical hyperspectral imaging (FT-NIR and FT-IR), and many others. Theoretical studies in quantum mechanics (QM), QCD, QG, AQFT, and quantum theories on a lattice (QTL) also employ Fourier transforms. \textbf{Fourier--Stieltjes transforms} and \textbf{measured groupoid transforms} are useful generalizations of the ordinary Fourier transform, as summarized in the following table. \subsubsection*{Fourier Transforms and Generalized FTs} \begin{center} \begin{tabular}{|p{0.16\textwidth}|p{0.20\textwidth}|p{0.20\textwidth}|p{0.18\textwidth}|p{0.18\textwidth}|} \hline $f(t)$ & $\mathcal{F}\{f(t)\}=\hat{f}(x)$ & Conditions & Explanation & Description \tabularnewline \hline Gaussian function & Gaussian function & general & In statistics and spectroscopy & Gaussian profiles remain Gaussian under Fourier transformation \tabularnewline \hline Lorentzian function & Exponential-type transform & general & In spectroscopy & Associated with exponentially decaying time-domain signals \tabularnewline \hline Step or rectangular function & $\sin(x)/x$-type function & general & FT of a rectangular pulse & Sinc-type transform \tabularnewline \hline Triangular function & $\sin^2(x)/x^2$-type function & general & Transform of a triangular profile & Squared-sinc-type transform \tabularnewline \hline Series of equidistant points & Periodic reciprocal-space series & general & Ideal periodic lattice & Used in diffraction theory \tabularnewline \hline Lattice of infinite planes & Series of equidistant reciprocal-space points & general & One-dimensional reciprocal space & Used in crystallography and diffraction theory \tabularnewline \hline Helix wrapped on a cylinder & Bessel functions or Bessel--Fourier series & general & Physical crystallography & Experimentally truncated to a finite number of Bessel terms \tabularnewline \hline $c$ & $(\sqrt{2\pi})^{-1}c$ & Convention-dependent & Constant input & Normalization depends on Fourier-transform convention \tabularnewline \hline $f(t)$ & $\displaystyle \int \hat{f}(x)\,\overline{t(x)}\,dx$ & $f(t)\in L^1(G_l)$, with $G_l$ a locally compact groupoid \cite{RW97}; the integral is defined using a left Haar measure on $G_l$ & Fourier--Stieltjes transform & $\hat{f}(x)\in C_0(\widehat{G_l})$ \tabularnewline \hline $\hat{m}(x)$ & $\displaystyle \check{m}(t)=\int e^{itx}\,d\hat{m}(x)$ & as above & Inverse Fourier--Stieltjes transform & $\check{m}(t)\in L^1(G_l)$ (\cite{PALT2k1}, \cite{PALT2k3}) \tabularnewline \hline $\hat{m}(x)$ & $\displaystyle \check{m}(t)=\int e^{itx}\,d\hat{m}(x)$ & When $G_l=\mathbb{R}$ and the integral exists & Usual inverse Fourier transform & $\check{m}(t)\in\mathbb{R}$ \tabularnewline \hline \end{tabular} \end{center} \emph{Note.} The hat on $\hat{f}(x)$ and $\widehat{G_l}$ denotes the transformed quantity or, in the latter case, the dual object. \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}