# Balian-Low

###### Theorem 1 (Balian-Low)

Suppose $g\in L^{2}(\mathbb{R})$ and $g_{m,n}(x)=e^{2\pi imx}g(x-n)$, where $m,n\in\mathbb{Z}$. If $\{g_{m,n}:m,n\in\mathbb{Z}\}$ is an orthonormal basis for $L^{2}(\mathbb{R})$, then either

 $\int_{-\infty}^{\infty}x^{2}|g(x)|^{2}\;dx=\infty\text{ or }\int_{-\infty}^{% \infty}\xi^{2}|\hat{g}(\xi)|^{2}\;d\xi=\infty.$
Title Balian-Low BalianLow 2013-03-22 15:35:12 2013-03-22 15:35:12 swiftset (1337) swiftset (1337) 4 swiftset (1337) Theorem msc 42C10