PaleyWienerTheorem 2005-07-27 13:48:52 2007-05-25 02:50:12 Theorem % 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 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 $L^2(\mathbb{R})$, then there exists a function $g(t)\in L^2(-\gamma, \gamma)$ such that $$ f(z) = \frac{1}{\sqrt{2\pi}}\int_{-\gamma}^{\gamma}g(t)e^{izt}dt$$ for all $z$.