generalized Fourier transform

Definition 0.1.

Given a positive definite, measurable functionMathworldPlanetmath f(x) on the interval (-,) there exists a monotone increasing, real-valued bounded function α(t) such that:

f(x)=eitxd(α(t)), (0.1)

for all x 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 α(t) is nondecreasing and bounded then the measurable function defined by the above integral is called the Fourier-Stieltjes transform of α(t), and it is continuous in addition to being positive definite.


  • 1 A. Ramsay and M. E. Walter, Fourier-Stieltjes algebras of locally compact groupoidsPlanetmathPlanetmath, 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