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%%% Primary Title: generalized Fourier and measured groupoid transforms
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\begin{document}
\subsection{Generalized Fourier transforms}
\textbf{Fourier-Stieltjes} transforms and \textbf{measured \htmladdnormallink{groupoid}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html}} transforms are useful generalizations of the (much simpler) Fourier transform, as concisely shown in the following table (see also \htmladdnormallink{Fourier transforms}{TableOfFourierTransforms} )
- for the purpose of direct comparison with the latter transform. Unlike the more general \htmladdnormallink{Fourier-Stieltjes transform}{http://planetphysics.us/encyclopedia/StieltjesTransform.html}, the Fourier transform exists if and only if the \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} to be transformed is Lebesgue integrable over the whole real axis for $t \in{\mathbb{R}}$, or over the entire ${\mathbb{C}}$ \htmladdnormallink{domain}{http://planetphysics.us/encyclopedia/Bijective.html} when $\check{m}(t)$ is a complex function.
\begin{definition} \textbf{Fourier-Stieltjes transform}.
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
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. 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 continuous in addition to being positive definite.
\end{definition}
\subsubsection*{FT and FT-Generalizations}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline\hline
$f(t)$ & $\F{f(t)} = \hat{f}(x)= (2 \pi)^{-1}\int{e{(-itx)}dx}$ & Conditions* & Explanation & Description \\
\hline
$e^{-t} \theta (t)$ & $\F{[f(t)]}(x) = (2 \pi)^{-1}\int{\theta (t)e{(it^2x)}dx}$ & from $-\infty$ to +$\infty$ & From $Mathematica^{TM**}$\\
\hline
$c$ & $(\sqrt{2 \pi})^{-1}c$ & & & \\
& & Notice on the next line the overline & bar ($\overline{}$) placed above $t(x)$& \\
\hline
$f(t)$ & $\int \hat{f}(x) \overline{t(x)}dx$ & $f(t)\in{L^1(G_l)}$, with $G_l$ a & Fourier-Stieltjes transform & $\hat{f}(x)\in{C_0(\hat{G_l})}$ \\
& & \htmladdnormallink{locally compact groupoid}{http://planetphysics.us/encyclopedia/LocallyCompactGroupoid.html} \cite{RW97}; & & \\
& & $\int $ is defined \emph{via} & & \\
& & a left \htmladdnormallink{Haar measure}{http://planetphysics.us/encyclopedia/HigherDimensionalQuantumAlgebroid.html} on $G_l$ & & \\
\hline
$\hat{m}(x)$ & $\check{m}(t)= \int e^{itx}d\hat{m}(x)$ & as above & Inverse Fourier-Stieltjes & $\check{m}(t) \in{L^1(G_l)}$, \\
& & & transform & (\cite{PALT2k1}, \cite{PALT2k3}). \\
\hline
$\hat{m}(x)$ & $\check{m}(t) = \int e^{itx}d\hat{m}(x)$ & When $G_l=\mathbb{R}$, and it exists & This is the usual & $\check{m}(t) \in{\mathbb{R}}$ \\
& & only when $\hat{m}(x)$ is & Inverse Fourier transform & \\
& & \emph{Lebesgue integrable} on & & \\
& & the entire real axis & & \\
\hline\hline
\end{tabular}
\end{center}
*Note the `slash hat' on $\hat{f}(x)$ and $\hat{G_l}$;
**Calculated numerically using
\htmladdnormallink{this link to $Mathematica^{TM}$}{http://demonstrations.wolfram.com/NumericalApproximationOfTheFourierTransformByTheFastFourierT/}
\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) \htmladdnormallink{Free PDF file download}{http://aux.planetmath.org/files/objects/10739/AFourierStjelties_LocallyCompactsGds_Harmonic0310138v1.pdf}.
\end{thebibliography}
\end{document}
