Balian-Low theorem

In mathematics, the Balian-Low theorem in Fourier analysis is named for Roger Balian and Francis Low.

g m, n (x) = e 2π i m x g (x - n),

for integers m and n. The Balian-Low theorem states that if

$\{g_{m,n}: m, n \in \mathbb{Z}\}$

$L^2(\mathbb{R}),$

then either

$\int_{-\infty}^\infty x^2 | g(x)|^2\; dx = \infty \quad \textrm{or} \quad \int_{-\infty}^\infty \xi^2|\hat{g}(\xi)|^2\; d\xi = \infty.$

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This article incorporates material from Balian-Low on PlanetMath, which is licensed under the GFDL.