PlanetPhysics/Table of Fourier and Generalized Transforms
\newcommand{\sqdiagram}[9]{Failed to parse (unknown function "\diagram"): {\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}} }
Table of Fourier and generalized Fourier transforms
[edit | edit source]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
[edit | edit source]References
[edit | edit source]- ↑ 1.0 1.1 A. Ramsay and M. E. Walter, Fourier-Stieltjes algebras of locally compact groupoids, J. Functional Anal . 148 : 314-367 (1997).
- ↑ 2.0 2.1 A. L. T. Paterson, The Fourier algebra for locally compact groupoids., Preprint, (2001).
- ↑ 3.0 3.1 A. L. T. Paterson, The Fourier-Stieltjes and Fourier algebras for locally compact groupoids., (2003) Free PDF file download.