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generalized Fourier transform (Definition)

Fourier-Stieltjes Transform

Definition 1.1   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:
$\displaystyle f(x)=\int_\mathbb{R}e^{itx}d(\alpha (t)),$ (1.1)

for all $x \in{\mathbb{R}}$ except a `small' set, that is a finite set which contains only a small number of values. 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: determination of Fourier coefficients, generalized Fourier and measured groupoid transforms, 2D-FT MR- Imaging and related Nobel awards, two-dimensional Fourier transforms

Other names:  Fourier-Stieltjes transform, Stieltjes transform, Fourier-Stieltjes integral
Also defines:  Fourier-Stieltjes transform, Fourier-Stieltjes algebra of a groupoid, Fourier-Stieltjes integral
Keywords:  Fourier-Stieltjes algebra of a groupoid

Cross-references: function, measurable function
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This is version 6 of generalized Fourier transform, born on 2009-01-11, modified 2009-04-19.
Object id is 373, canonical name is GeneralizedFourierTransform.
Accessed 1650 times total.

Classification:
Physics Classification: 02. (Mathematical methods in physics)
 02.30.Nw (Fourier analysis)
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