PlanetPhysics/Generalized Fourier and Measured Groupoid Transforms

\newcommand{\sqdiagram}[9]{Failed to parse (unknown function "\diagram"): {\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}} }

Generalized Fourier transforms

Fourier-Stieltjes transforms and measured [[../QuantumOperatorAlgebra5/|groupoid]] transforms are useful generalizations of the (much simpler) Fourier transform, as concisely shown in the following table (see also [TableOfFourierTransforms Fourier transforms] ) - for the purpose of direct comparison with the latter transform. Unlike the more general [[../StieltjesTransform/|Fourier-Stieltjes transform]], the Fourier transform exists if and only if the [[../Bijective/|function]] to be transformed is Lebesgue integrable over the whole real axis for $t\in ℝ$, or over the entire $ℂ$ [[../Bijective/|domain]] when $\check{m}(t)$ is a complex function.

Fourier-Stieltjes transform .

Given a positive definite, [[../LebesgueMeasure/|measurable function]] $f(x)$ on the interval $(-\mathrm{\infty },\mathrm{\infty })$ there exists a monotone increasing, real-valued bounded function $\alpha (t)$ such that:

$f(x)=\underset{ℝ}{\int }{e}^{itx}d(\alpha (t),$

for all $x\in ℝ$ except a small set. When $f(x)$ is defined as above and if $\alpha (t)$ is nondecreasing and bounded then the measurable function defined by the above integral is called the Fourier-Stieltjes transform of $\alpha (t)$, and it is continuous in addition to being positive definite.

\subsubsection*{FT and FT-Generalizations}

$f(t)$	$ℱf(t)=\hat{f}(x)=(2\pi {)}^{-1}\int e(-itx)dx$	Conditions*	Explanation	Description
$e^{-t}\theta (t)$	$ℱ[f(t)](x)=(2\pi {)}^{-1}\int \theta (t)e(i{t}^{2}x)dx$	from $-\mathrm{\infty }$ to +$\mathrm{\infty }$	From $Mathematic{a}^{TM**}$
$c$	$(\sqrt{2\pi }{)}^{-1}c$
		Notice on the next line the overline	bar ($\stackrel{}{‾}$) placed above $t(x)$
$f(t)$	$\int \hat{f}(x)\overline{t(x)}dx$	$f(t)\in L^1(G_l)$, with $G_l$ a	Fourier-Stieltjes transform	$\hat{f}(x)\in C_0(\widehat{G_l})$
		[[../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 $\widehat{G_l}$;
  • Calculated numerically using this link to $Mathematic{a}^{TM}$