# table of generalized Fourier and measured groupoid transforms

## 0.1 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- with the same format as C. Woo’s Feature on Fourier transforms (http://planetmath.org/TableOfFourierTransforms) - 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 Generalizations

Title table of generalized Fourier and measured groupoid transforms TableOfGeneralizedFourierAndMeasuredGroupoidTransforms 2013-03-22 18:10:27 2013-03-22 18:10:27 bci1 (20947) bci1 (20947) 46 bci1 (20947) Topic msc 55U99 Fourier-Stieltjes transforms FourierTransform TwoDimensionalFourierTransforms Fourier-Stieltjes and measured groupoid transforms