| Version 41 |
Version 40 |
| %%Generalized FT |
%%Generalized FT |
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| \textbf{generalized Fourier transforms} |
\textbf{generalized Fourier transforms} |
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| \textbf{Fourier-Stieltjes} transforms and \textbf{measured groupoid} transforms are useful generalizations of the (much simpler) Fourier transform, as concisely shown in the following table- |
\textbf{Fourier-Stieltjes} transforms and \textbf{measured groupoid} transforms are useful generalizations of the (much simpler) Fourier transform, as concisely shown in the following table- |
| with the same format as CWoo's Feature on \PMlinkname{Fourier transforms}{TableOfFourierTransforms} |
with the same format as CWoo's Feature on \PMlinkname{Fourier transforms}{TableOfFourierTransforms} |
| - for the purpose of direct comparison with the latter transform. Unlike the more general Fourier-Stieltjes |
- for the purpose of direct comparison with the latter transform. Unlike the more general Fourier-Stieltjes |
| transform, the Fourier transform exists if and only if the function to be transformed is Lebesgue integrable over the whole real axis for $t \in{\mathbb{R}}$, or over the entire ${\mathbb{C}}$ domain when $\check{m}(t)$ is a complex function. |
transform, the Fourier transform exists if and only if the function to be transformed is Lebesgue integrable over the whole real axis for $t \in{\mathbb{R}}$, or over the entire ${\mathbb{C}}$ domain when $\check{m}(t)$ is a complex function. |
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| \begin{definition} \textbf{Fourier-Stieltjes transform}. |
\begin{definition} \textbf{Fourier-Stieltjes transform}. |
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| Given a \emph{positive definite, measurable function} $f(x)$ on the interval |
Given a \emph{positive definite, measurable function} $f(x)$ on the interval |
| $(-\infty ,\infty)$ there exists a monotone increasing, real-valued bounded |
$(-\infty ,\infty)$ there exists a monotone increasing, real-valued bounded |
| function $ \alpha (t)$ such that: |
function $ \alpha (t)$ such that: |
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|
| \begin{equation} |
\begin{equation} |
| f(x)=\int_\mathbb{R}e^{itx}d(\alpha (t), |
f(x)=\int_\mathbb{R}e^{itx}d(\alpha (t), |
| \end{equation} |
\end{equation} |
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| 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 \emph{continuous} in addition to being \emph{positive definite}. |
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 \emph{continuous} in addition to being \emph{positive definite}. |
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| \end{definition} |
\end{definition} |
|
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| \subsubsection*{\textbf{FT} \textbf{Extensions or Generalizations}} |
\subsubsection*{\textbf{FT} \textbf{Extensions or Generalizations}} |
| \begin{center} |
\begin{center} |
| \begin{tabular}{|c|c|c|c|c|} |
\begin{tabular}{|c|c|c|c|c|} |
| \hline\hline |
\hline\hline |
| $f(t)$ & $\F{f(t)} = \hat{f}(x)$ & Conditions* & Explanation & Description \\ |
$f(t)$ & $\F{f(t)} = \hat{f}(x)$ & Conditions* & Explanation & Description \\ |
| \hline |
\hline |
| $c$ & $(\sqrt{2 \pi})^{-1}c$ & Notice on the next line the & & \\ |
$c$ & $(\sqrt{2 \pi})^{-1}c$ & Notice on the next line the & & \\ |
| & & overline bar placed above $t(x)$ & & \\ |
& & overline bar placed above $t(x)$ & & \\ |
| \hline |
\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 & $\hat{f}(x)\in{C_0(\hat{G_l})}$ \\ |
$f(t)$ & $\int \hat{f}(x) \overline{t(x)}dx$ & $f(t)\in{L^1(G_l)}$, with $G_l$ a & Fourier-Stieltjes transform & $\hat{f}(x)\in{C_0(\hat{G_l})}$ \\ |
| & & locally compact groupoid \cite{RW97}; & & \\ |
& & locally compact groupoid \cite{RW97}; & & \\ |
| & & $\int $ is defined \emph{via} & & \\ |
& & $\int $ is defined \emph{via} & & \\ |
| & & a left Haar measure on $G_l$ & & \\ |
& & a left Haar measure on $G_l$ & & \\ |
| \hline |
\hline |
| $\hat{m}(x)$ & $\check{m}(t)= \int e^{itx}d\hat{m}(x)$ & as above & Inverse Fourier-Stieltjes & $\check{m}(t) \in{L^1(G_l)}$, \\ |
$\hat{m}(x)$ & $\check{m}(t)= \int e^{itx}d\hat{m}(x)$ & as above & Inverse Fourier-Stieltjes & $\check{m}(t) \in{L^1(G_l)}$, \\ |
| & & & transform & (\cite{PALT2k1}, \cite{PALT2k3}). \\ |
& & & transform & (\cite{PALT2k1}, \cite{PALT2k3}). \\ |
| \hline |
\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 & $\check{m}(t) \in{\mathbb{R}}$ \\ |
$\hat{m}(x)$ & $\check{m}(t) = \int e^{itx}d\hat{m}(x)$ & 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 & \\ |
& & only when $\hat{m}(x)$ is & Inverse Fourier transform & \\ |
| & & \emph{Lebesgue integrable} on & & \\ |
& & \emph{Lebesgue integrable} on & & \\ |
| & & the entire real axis & & \\ |
& & the entire real axis & & \\ |
| \hline\hline |
\hline\hline |
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|
|
| \end{tabular} |
\end{tabular} |
| \end{center} |
\end{center} |
| *Note the `slash hat' on $\hat{f}(x)$ and $\hat{G_l}$. |
*Note the `slash hat' on $\hat{f}(x)$ and $\hat{G_l}$. |
|
|
| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{RW97} |
\bibitem{RW97} |
| A. Ramsay and M. E. Walter, Fourier-Stieltjes algebras of locally compact groupoids, |
A. Ramsay and M. E. Walter, Fourier-Stieltjes algebras of locally compact groupoids, |
| \emph{J. Functional Anal}. \textbf{148}: 314-367 (1997). |
\emph{J. Functional Anal}. \textbf{148}: 314-367 (1997). |
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| \bibitem{PALT2k1} |
\bibitem{PALT2k1} |
| A. L. T. Paterson, The Fourier algebra for locally compact groupoids., Preprint, (2001). |
A. L. T. Paterson, The Fourier algebra for locally compact groupoids., Preprint, (2001). |
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|
| \bibitem{PALT2k3} |
\bibitem{PALT2k3} |
| A. L. T. Paterson, The Fourier-Stieltjes and Fourier algebras for locally |
A. L. T. Paterson, The Fourier-Stieltjes and Fourier algebras for locally |
| compact groupoids., (2003) \PMlinkexternal{Free PDF file download}{http://aux.planetmath.org/files/objects/10739/AFourierStjelties_LocallyCompactsGds_Harmonic0310138v1.pdf}. |
compact groupoids., (2003) \PMlinkexternal{Free PDF file download}{http://aux.planetmath.org/files/objects/10739/AFourierStjelties-LocallyCompactsGds-Harmonic0310138v1.pdf} |
|
|
| \end{thebibliography} |
\end{thebibliography} |
