generalized Fourier transform

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 boundedfunction $\\alpha(t)$ 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 locallycompact groupoids, (2003).