Talk:PlanetPhysics/Generalized Fourier Transform

Original TeX Content from PlanetPhysics Archive

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\begin{document}

 \section{Fourier-Stieltjes Transform}
\begin{definition}
Given a \emph{positive definite, \htmladdnormallink{measurable function}{http://planetphysics.us/encyclopedia/LebesgueMeasure.html}} $f(x)$ on the interval
$(-\infty ,\infty)$ there exists a monotone increasing, real-valued bounded
\htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} $ \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} 

\end{document}