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Viewing Version
7
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'Paley-Wiener theorem'
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| Title of object: |
Paley-Wiener theorem |
| Canonical Name: |
PaleyWienerTheorem |
| Type: |
Theorem |
| Created on: |
2005-07-27 13:48:52 |
| Modified on: |
2007-05-23 19:53:12 |
| Classification: |
msc:30E99 |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here |
Content:
Let $f(z)$ be an entire function such that $\vert f(z)\vert \leq K
e^{\gamma \vert z\vert}$ for some $K \geq 0$ and
$\gamma > 0$. If the restriction of $f$ to the real line
is in \PMlinkname{$L^2(\mathbb{R})$}{LpSpace} , then there exists a function
$g(t)\in$
\PMlinkname{$L^2(-\gamma, \gamma)$}{LpSpace} such that
$$ f(z) = \frac{1}{\sqrt{2\pi}}\int_{-\gamma}^{\gamma}g(t)e^{izt}dt$$
for all $z$. |
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