Balian-Low theorem

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

Suppose g is a square-integrable function on the real line, and consider the so-called Gabor system

gm,n(x) = e2πimxg(x − n),

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

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

is an orthonormal basis for the Hilbert space

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.

The The Balian-Low theorem has been extended to exact Gabor frames.

[edit] References

  • John J. Benedetto, Christopher Heil, and David F. Walnut (1994). "Differentiation and the Balian-Low Theorem". Journal of Fourier Analysis and Applications Volume 1, Number 4: 355–402. doi:10.1007/s00041-001-4016-5. 

[edit] See also

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

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