generalized Fourier transform
Definition 0.1.
Given a positive definite, measurable function^{} $f(x)$ on the interval $(-\mathrm{\infty},\mathrm{\infty})$ there exists a monotone increasing, real-valued bounded function $\alpha (t)$ such that:
$$f(x)={\int}_{\mathbb{R}}{e}^{itx}d(\alpha (t)),$$ | (0.1) |
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.
References
- 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).
Title | generalized Fourier transform |
Canonical name | GeneralizedFourierTransform |
Date of creation | 2013-03-22 18:16:07 |
Last modified on | 2013-03-22 18:16:07 |
Owner | bci1 (20947) |
Last modified by | bci1 (20947) |
Numerical id | 14 |
Author | bci1 (20947) |
Entry type | Definition |
Classification | msc 55P99 |
Classification | msc 55R10 |
Classification | msc 55R65 |
Classification | msc 55R37 |
Classification | msc 42B10 |
Classification | msc 42A38 |
Synonym | Stieltjes-Fourier transform |
Related topic | FourierStieltjesAlgebraOfAGroupoid |
Related topic | TwoDimensionalFourierTransforms |
Related topic | DiscreteFourierTransform |
Defines | positive definite- measurable function |