# PlanetPhysics/Generalized Fourier and Measured Groupoid Transforms

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### Generalized 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 (see also [TableOfFourierTransforms Fourier transforms] ) - for the purpose of direct comparison with the latter transform. Unlike the more general Fourier-Stieltjes transform, the Fourier transform exists if and only if the function to be transformed is Lebesgue integrable over the whole real axis for $t\in {\mathbb {R} }$ , or over the entire ${\mathbb {C} }$ domain when ${\check {m}}(t)$ is a complex function.

Fourier-Stieltjes transform .

Given a positive definite, measurable function $f(x)$ on the interval $(-\infty ,\infty )$ there exists a monotone increasing, real-valued bounded function $\alpha (t)$ such that:

$f(x)=\int _{\mathbb {R} }e^{itx}d(\alpha (t),$ for all $x\in {\mathbb {R} }$ 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}

 $f(t)$ ${\mathcal {F}}{f(t)}={\hat {f}}(x)=(2\pi )^{-1}\int {e{(-itx)}dx}$ Conditions* Explanation Description $e^{-t}\theta (t)$ ${\mathcal {F}}{[f(t)]}(x)=(2\pi )^{-1}\int {\theta (t)e{(it^{2}x)}dx}$ from $-\infty$ to +$\infty$ From $Mathematica^{TM**}$ $c$ $({\sqrt {2\pi }})^{-1}c$ Notice on the next line the overline bar (${\overline {}}$ ) 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 ${\hat {f}}(x)\in {C_{0}({\hat {G_{l}}})}$ 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