PlanetPhysics/Generalized Fourier and Measured Groupoid Transforms

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Generalized Fourier transformsEdit

Fourier-Stieltjes transforms and measured groupoid transforms are useful generalizations of the (much simpler) Fourier transform, as concisely shown in the following table (see also [TableOfFourierTransforms 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 integrable over the whole real axis for , or over the entire domain when is a complex function.

Fourier-Stieltjes transform .

Given a positive definite, measurable function on the interval there exists a monotone increasing, real-valued bounded function such that:

for all except a small set. When is defined as above and if is nondecreasing and bounded then the measurable function defined by the above integral is called the Fourier-Stieltjes transform of , and it is continuous in addition to being positive definite.

\subsubsection*{FT and FT-Generalizations}

Conditions* Explanation Description
from to + From
Notice on the next line the overline bar () placed above
, with a Fourier-Stieltjes transform
locally compact groupoid [1];
is defined via
a left Haar measure on
as above Inverse Fourier-Stieltjes ,
transform ([2], [3]).
When , and it exists This is the usual
only when is Inverse Fourier transform
Lebesgue integrable on
the entire real axis

All SourcesEdit



  1. 1.0 1.1 A. Ramsay and M. E. Walter, Fourier-Stieltjes algebras of locally compact groupoids, J. Functional Anal . 148 : 314-367 (1997).
  2. 2.0 2.1 A. L. T. Paterson, The Fourier algebra for locally compact groupoids., Preprint, (2001).
  3. 3.0 3.1 A. L. T. Paterson, The Fourier-Stieltjes and Fourier algebras for locally compact groupoids., (2003) Free PDF file download.