1 Fourier-Stieltjes Transform
Definition 1.1. Given a positive definite, measurable function f(x) on the interval (−∞,∞)
there exists a monotone increasing, real-valued bounded function α(t) such that:
for all x ∈ 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 α(t) is nondecreasing and bounded then the measurable
function defined by the above integral is called the Fourier-Stieltjes transform of α(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).
