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][2][3]

ReferencesEdit

  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.