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