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 transformMathworldPlanetmath, as concisely shown in the following table- with the same format as C. Woo’s Feature on 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 integrableMathworldPlanetmath over the whole real axis for t, or over the entire domain when mˇ(t) is a complex function.

Definition 0.1.

Fourier-Stieltjes transform.

Given a positive definitePlanetmathPlanetmath, 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. 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 continuousMathworldPlanetmath in addition to being positive definite.

FT Generalizations

Title table of generalized Fourier and measured groupoid transforms
Canonical name TableOfGeneralizedFourierAndMeasuredGroupoidTransforms
Date of creation 2013-03-22 18:10:27
Last modified on 2013-03-22 18:10:27
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 46
Author bci1 (20947)
Entry type Topic
Classification msc 55U99
Synonym Fourier-Stieltjes transforms
Related topic FourierTransform
Related topic TwoDimensionalFourierTransforms
Defines Fourier-Stieltjes and measured groupoid transforms