PlanetPhysics/Generalized Fourier and Measured Groupoid Transforms
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Generalized Fourier transforms
editFourier-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 |
- Note the 'slash hat' on and ;
- Calculated numerically using this link to
All Sources
editReferences
edit- ↑ 1.0 1.1 A. Ramsay and M. E. Walter, Fourier-Stieltjes algebras of locally compact groupoids, J. Functional Anal . 148 : 314-367 (1997).
- ↑ 2.0 2.1 A. L. T. Paterson, The Fourier algebra for locally compact groupoids., Preprint, (2001).
- ↑ 3.0 3.1 A. L. T. Paterson, The Fourier-Stieltjes and Fourier algebras for locally compact groupoids., (2003) Free PDF file download.