# 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 |

- Note the 'slash hat' on and ;
- Calculated numerically using this link to

## All SourcesEdit

^{[1]}^{[2]}^{[3]}

## ReferencesEdit

- ↑
^{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.