BalianLow 2005-11-22 13:45:29 2005-11-22 13:45:29 Theorem OrthonormalBasis % 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 \newtheorem{thm}{Theorem} \newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \begin{thm}[Balian-Low] Suppose $g \in L^2(\R)$ and $g_{m,n}(x) = e^{2\pi i m x} g(x - n)$, where $m,n \in \Z$. If $\{g_{m,n}: m, n \in \Z\}$ is an orthonormal basis for $L^2(\R)$, then either $$\int_{-\infty}^\infty x^2 |g(x)|^2\; dx = \infty \text{ or } \int_{-\infty}^\infty \xi^2|\hat{g}(\xi)|^2\; d\xi = \infty. $$ \end{thm}