generalized Fourier transform

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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