PlanetPhysics/Table of Fourier and Generalized Transforms

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Table of Fourier and generalized Fourier transforms

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}

$f(t)$	${\mathcal {F}}{f(t)}={\hat {f}}(x)$	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	$sin(x)/x$	general	FT of a square wave	`slit' function
sawtooth function	$sin^{2}(x)/x^{2}$	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
$c$	$({\sqrt {2\pi }})^{-1}c$	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	Fourier-Stieltjes transform	${\hat {f}}(x)\in {C_{0}({\hat {G_{l}}})}$
		locally compact groupoid ;
		$\int$ is defined via
		a left 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 $G_{l}=\mathbb {R}$, and it exists	This is the usual	${\check {m}}(t)\in {\mathbb {R} }$
		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 ${\hat {G_{l}}}$.