# PlanetPhysics/Table of Fourier and Generalized Transforms

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### 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 |
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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 |
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the entire real axis |

*Note the `slash hat' on and .

## All Sources

[edit | edit source]^{[1]}
^{[2]}
^{[3]}

## 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.