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Theorem 1 (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 $$\int_{-\infty}^\infty x^2 |g(x)|^2\; dx = \infty \text{ or } \int_{-\infty}^\infty \xi^2|\hat{g}(\xi)|^2\; d\xi = \infty. $$
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"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 2225 times total.
Classification:
| AMS MSC: | 42C10 (Fourier analysis :: Nontrigonometric Fourier analysis :: Fourier series in special orthogonal functions ) |
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Pending Errata and Addenda
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