# 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 ${\displaystyle t\in {\mathbb {R} }}$, or over the entire ${\displaystyle {\mathbb {C} }}$ domain when ${\displaystyle {\check {m}}(t)}$ is a complex function.

Fourier-Stieltjes transform .

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

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

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

\subsubsection*{FT and FT-Generalizations}

 ${\displaystyle f(t)}$ ${\displaystyle {\mathcal {F}}{f(t)}={\hat {f}}(x)=(2\pi )^{-1}\int {e{(-itx)}dx}}$ Conditions* Explanation Description ${\displaystyle e^{-t}\theta (t)}$ ${\displaystyle {\mathcal {F}}{[f(t)]}(x)=(2\pi )^{-1}\int {\theta (t)e{(it^{2}x)}dx}}$ from ${\displaystyle -\infty }$ to +${\displaystyle \infty }$ From ${\displaystyle Mathematica^{TM**}}$ ${\displaystyle c}$ ${\displaystyle ({\sqrt {2\pi }})^{-1}c}$ Notice on the next line the overline bar (${\displaystyle {\overline {}}}$) placed above ${\displaystyle t(x)}$ ${\displaystyle f(t)}$ ${\displaystyle \int {\hat {f}}(x){\overline {t(x)}}dx}$ ${\displaystyle f(t)\in {L^{1}(G_{l})}}$, with ${\displaystyle G_{l}}$ a Fourier-Stieltjes transform ${\displaystyle {\hat {f}}(x)\in {C_{0}({\hat {G_{l}}})}}$ locally compact groupoid [1]; ${\displaystyle \int }$ is defined via a left Haar measure on ${\displaystyle G_{l}}$ ${\displaystyle {\hat {m}}(x)}$ ${\displaystyle {\check {m}}(t)=\int e^{itx}d{\hat {m}}(x)}$ as above Inverse Fourier-Stieltjes ${\displaystyle {\check {m}}(t)\in {L^{1}(G_{l})}}$, transform ([2], [3]). ${\displaystyle {\hat {m}}(x)}$ ${\displaystyle {\check {m}}(t)=\int e^{itx}d{\hat {m}}(x)}$ When ${\displaystyle G_{l}=\mathbb {R} }$, and it exists This is the usual ${\displaystyle {\check {m}}(t)\in {\mathbb {R} }}$ only when ${\displaystyle {\hat {m}}(x)}$ is Inverse Fourier transform Lebesgue integrable on the entire real axis

## References

1. A. Ramsay and M. E. Walter, Fourier-Stieltjes algebras of locally compact groupoids, J. Functional Anal . 148 : 314-367 (1997).
2. A. L. T. Paterson, The Fourier algebra for locally compact groupoids., Preprint, (2001).
3. A. L. T. Paterson, The Fourier-Stieltjes and Fourier algebras for locally compact groupoids., (2003) Free PDF file download.