PlanetMath: Balian-Low

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Balian-Low

(Theorem)

Theorem 1 (Balian-Low) Suppose $g \in L^2(\mathbb{R})$ and $g_{m,n}(x) = e^{2\pi i m x} g(x - n)$ , 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

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

"Balian-Low" is owned by swiftset.

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Keywords:

OrthonormalBasis

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Cross-references: orthonormal basis

This is version 1 of Balian-Low, born on 2005-11-22.
Object id is 7497, canonical name is BalianLow.
Accessed 1577 times total.

Classification:

AMS MSC:

42C10 (Fourier analysis :: Nontrigonometric Fourier analysis :: Fourier series in special orthogonal functions )

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