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generalized Fourier transform (Definition)
Definition 0.1   Fourier-Stieltjes transform. Given a positive definite, measurable function $ f(x)$ on the interval $ (-\infty ,\infty)$ there exists a monotone increasing, real-valued bounded function $ \alpha (t)$ such that:
$\displaystyle f(x)=\int_\mathbb{R}e^{itx}d(\alpha (t),$ (0.1)

for all $ x \in{\mathbb{R}}$ except a small set. When $ f(x)$ is defined as above and if $ \alpha(t)$ is nondecreasing and bounded then the measurable function defined by the above integral is called the Fourier-Stieltjes transform of $ \alpha(t)$, and it is continuous in addition to being positive definite.

Bibliography

1
A. Ramsay and M. E. Walter, Fourier-Stieltjes algebras of locally compact groupoids, J. Functional Anal. 148: 314-367 (1997).
2
A. L. T. Paterson, The Fourier algebra for locally compact groupoids., Preprint, (2001).
3
A. L. T. Paterson, The Fourier-Stieltjes and Fourier algebras for locally compact groupoids, (2003).



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See Also: Fourier-Stieltjes algebra of a groupoid, two-dimensional Fourier transforms, discrete Fourier transform

Other names:  Stieltjes-Fourier transform
Also defines:  measurable function
Keywords:  Fourier-Stieltjes transform, harmonic and anharmonic analysis
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Cross-references: addition, continuous, integral, bounded, bounded function, monotone increasing, interval, positive definite, Fourier-Stieltjes transform
There are 9 references to this entry.

This is version 7 of generalized Fourier transform, born on 2008-07-25, modified 2008-09-07.
Object id is 10871, canonical name is FourierStieltjesTransform.
Accessed 570 times total.

Classification:
AMS MSC: 42A38 (Fourier analysis :: Fourier analysis in one variable :: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type)
 42B10 (Fourier analysis :: Fourier analysis in several variables :: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type)
 55R37 (Algebraic topology :: Fiber spaces and bundles :: Maps between classifying spaces)
 55R65 (Algebraic topology :: Fiber spaces and bundles :: Generalizations of fiber spaces and bundles)
 55R10 (Algebraic topology :: Fiber spaces and bundles :: Fiber bundles)
 55P99 (Algebraic topology :: Homotopy theory :: Miscellaneous)
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