# generalized Fourier transform

###### Definition 0.1.

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, that is a finite set which contains only a small number of values. 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.

## 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).
 Title generalized Fourier transform Canonical name GeneralizedFourierTransform Date of creation 2013-03-22 18:16:07 Last modified on 2013-03-22 18:16:07 Owner bci1 (20947) Last modified by bci1 (20947) Numerical id 14 Author bci1 (20947) Entry type Definition Classification msc 55P99 Classification msc 55R10 Classification msc 55R65 Classification msc 55R37 Classification msc 42B10 Classification msc 42A38 Synonym Stieltjes-Fourier transform Related topic FourierStieltjesAlgebraOfAGroupoid Related topic TwoDimensionalFourierTransforms Related topic DiscreteFourierTransform Defines positive definite- measurable function