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table of Fourier and generalized transforms (Data Structure)

Table of Fourier and generalized Fourier transforms

Fourier transforms 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., X-ray, neutron, electron diffraction), chemical Hyperspectral Imaging (FT-NIR, FT-IR), and so on. Theoretical studies in quantum mechanics (QM), QCD, QG, AQFT, quantum theories on a lattice (QTL) also employ Fourier transforms.

Fourier-Stieltjes transforms and measured groupoid transforms are useful generalizations of the (much simpler) Fourier transform, as concisely shown in the following table.

Fourier Transforms and Generalized FTs

$f(t)$ $\mathcal F{f(t)} = \hat{f}(x)$ Conditions* Explanation Description
Gaussian function Gaussian function general In statistics, and also in spectroscopy
Lorentzian function Lorentzian function general In spectroscopy experimentally truncated to the single exponential function with a negative exponent
step function $sin(x)/x$ general FT of a square wave `slit' function
sawtooth function $sin^2(x)/x^2$ general a triangle zero baseline
series of equidistant points .... (inf.) group of equidistant planes general lattice of infinite planes used in diffraction theory
lattice of infinite planes, (or 1D paracrystal) series of equidistant points .... general one-dimensional reciprocal space used in crystallography/diffraction theory
Helix wrapped on a cylinder Bessel functions/ series general In Physical Crystallography experimentally truncated to the first (finite) n-th order Bessel functions
$c$ $(\sqrt{2 \pi})^{-1}c$ Notice on the next line the overline bar placed above $t(x)$ general Integration constant
$f(t)$ $\int \hat{f}(x) \overline{t(x)}dx$ $f(t)\in{L^1(G_l)}$, with $G_l$ a Fourier-Stieltjes transform $\hat{f}(x)\in{C_0(\hat{G_l})}$
    locally compact groupoid [1];    
    $\int $ is defined via    
    a left Haar measure on $G_l$    
$\hat{m}(x)$ $\check{m}(t)= \int e^{itx}d\hat{m}(x)$ as above Inverse Fourier-Stieltjes $\check{m}(t) \in{L^1(G_l)}$,
      transform ([2], [3]).
$\hat{m}(x)$ $\check{m}(t) = \int e^{itx}d\hat{m}(x)$ 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  
    Lebesgue integrable on    
    the entire real axis    
*Note the `slash hat' on $\hat{f}(x)$ and $\hat{G_l}$.

Bibliography

1
A. Ramsay and M. E. Walter, Fourier-Stieltjes algebras of locally compact groupoids, J. Functional Anal. 148: 314-367 (1997).
2
A. L. T. Paterson, The Fourier algebra for locally compact groupoids., Preprint, (2001).
3
A. L. T. Paterson, The Fourier-Stieltjes and Fourier algebras for locally compact groupoids., (2003) Free PDF file download.



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See Also: generalized Fourier and measured groupoid transforms, table of Laplace transforms, Bessel functions and diffraction by helical structures

Other names:  FT, FFT
Also defines:  quantum theories on a lattice, QM, QTL, FT-NIR, FT-IR
Keywords:  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

Cross-references: Haar measure, locally compact groupoid, Fourier-Stieltjes transform, Bessel functions, group, wave, square, Lorentzian, function, groupoid, quantum theories, AQFT, QG, QCD, quantum mechanics, Hyperspectral Imaging, neutron, Fourier transforms
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This is version 22 of table of Fourier and generalized transforms, born on 2009-04-22, modified 2009-05-02.
Object id is 685, canonical name is TableOfFourierAndGeneralizedTransforms.
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Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
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