Fourier-Stieltjes transforms and measured groupoid transforms are useful generalizations of the (much simpler) Fourier transform, as concisely shown in the following table.
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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 |
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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 |
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Notice on the next line the overline bar placed above ![]() |
general | Integration constant |
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Fourier-Stieltjes transform |
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locally compact groupoid [1]; | ||||
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a left Haar measure
on ![]() |
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as above | Inverse Fourier-Stieltjes |
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transform | ([2], [3]). | |||
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When
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This is the usual |
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only when
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Inverse Fourier transform | |||
Lebesgue integrable on | ||||
the entire real axis |
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