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PlanetPhysics/Table of Fourier and Generalized Transforms

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Table of Fourier and generalized Fourier transforms

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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.

\subsubsection*{Fourier Transforms and Generalized FTs}

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 general FT of a square wave `slit' function
sawtooth function 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

Notice on the next line the overline bar placed above general Integration constant
, with a Fourier-Stieltjes transform
locally compact groupoid [1];
is defined via
a left Haar measure on
as above Inverse Fourier-Stieltjes ,
transform ([2], [3]).
When , and it exists This is the usual
only when is Inverse Fourier transform
Lebesgue integrable on
the entire real axis

*Note the `slash hat' on and .

All Sources

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[1] [2] [3]

References

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  1. 1.0 1.1 A. Ramsay and M. E. Walter, Fourier-Stieltjes algebras of locally compact groupoids, J. Functional Anal . 148 : 314-367 (1997).
  2. 2.0 2.1 A. L. T. Paterson, The Fourier algebra for locally compact groupoids., Preprint, (2001).
  3. 3.0 3.1 A. L. T. Paterson, The Fourier-Stieltjes and Fourier algebras for locally compact groupoids., (2003) Free PDF file download.