generalized Fourier transformGeneralizedFourierTransform2009-01-11 03:19:452009-04-19 14:23:14DefinitionFourier-Stieltjes transformFourier-Stieltjes algebra of a groupoid% this is the default PlanetPhysics preamble. as your knowledge
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\def\D{\mathsf{D}}\section{Fourier-Stieltjes Transform}
\begin{definition}
Given a \emph{positive definite, measurable function} $f(x)$ on the interval
$(-\infty ,\infty)$ there exists a monotone increasing, real-valued bounded
function $ \alpha (t)$ such that:
\begin{equation}
f(x)=\int_\mathbb{R}e^{itx}d(\alpha (t)),
\end{equation}
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 \emph{the Fourier-Stieltjes transform of} $\alpha(t)$, and it is \emph{continuous} in addition to being \emph{positive definite}.
\end{definition}
\begin{thebibliography}{9}
\bibitem{RW97}
A. Ramsay and M. E. Walter, Fourier-Stieltjes algebras of locally compact groupoids,
\emph{J. Functional Anal}. \textbf{148}: 314-367 (1997).
\bibitem{PALT2k1}
A. L. T. Paterson, The Fourier algebra for locally compact groupoids., Preprint, (2001).
\bibitem{PALT2k3}
A. L. T. Paterson, The Fourier-Stieltjes and Fourier algebras for locally
compact groupoids, (2003).
\end{thebibliography}