generalized Fourier and measured groupoid transforms

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 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.

Definition 0.1   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:

$$f(x)=\int_\mathbb{R}e^{itx}d(\alpha (t),$$ (0.1)

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.

FT and FT-Generalizations

$f(t)$	$\mathcal F{f(t)} = \hat{f}(x)= (2 \pi)^{-1}\int{e{(-itx)}dx}$	Conditions*	Explanation	Description
$e^{-t} \theta (t)$	$\mathcal F{[f(t)]}(x) = (2 \pi)^{-1}\int{\theta (t)e{(it^2x)}dx}$	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 [1];
		$\int$ 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	([2], [3]).
$\hat{m}(x)$	$\check{m}(t) = \int e^{itx}d\hat{m}(x)$	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 $\hat{G_l}$ ; **Calculated numerically using this link to $Mathematica^{TM}$

Bibliography

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.

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• bci1

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