PlanetPhysics/Table of Fourier and Generalized Transforms

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Table of Fourier and generalized Fourier transforms

[[../FourierTransforms/|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., [[../FluorescenceCrossCorrelationSpectroscopy/|X-ray]], [[../Pions/|neutron]], electron diffraction), chemical [[../SpectralImaging/|Hyperspectral Imaging]] (FT-NIR, FT-IR), and so on. Theoretical studies in [[../QuantumParadox/|quantum mechanics]] (QM ), [[../LQG2/|QCD]], [[../LQG2/|QG]], [[../MetaTheorems/|AQFT]], [[../SpaceTimeQuantizationInQuantumGravityTheories/|quantum theories]] on a lattice (QTL ) also employ Fourier transforms.

Fourier-Stieltjes transforms and measured [[../GroupoidHomomorphism2/|groupoid]] transforms are useful generalizations of the (much simpler) Fourier transform, as concisely shown in the following table.

\subsubsection*{Fourier Transforms and Generalized FTs}

$f(t)$	$ℱf(t)=\hat{f}(x)$	Conditions*	Explanation	Description
Gaussian [[../Bijective/|function]]	Gaussian function	general	In statistics,	and also in spectroscopy
[[../LebesgueMeasure/|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 [[../PiecewiseLinear/|square]] [[../CosmologicalConstant/|wave]]	`slit' function
sawtooth function	$si{n}^2(x)/x^2$	general	a triangle	zero baseline
series of equidistant points ....	(inf.) [[../TrivialGroupoid/|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	[[../BesselEquation2/|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	[[../StieltjesTransform/|Fourier-Stieltjes transform]]	$\hat{f}(x)\in C_0(\widehat{G_l})$
		[[../LocallyCompactGroupoid/|locally compact groupoid]] ;
		$\int$ is defined via
		a left [[../HigherDimensionalQuantumAlgebroid/|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	(, ).
$\hat{m}(x)$	$\check{m}(t)=\int e^{itx}d\hat{m}(x)$	When $G_l=ℝ$, and it exists	This is the usual	$\check{m}(t)\in ℝ$
		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

$\widehat{G_l}$

.