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 [[Dennis Gabor|Gabor system]]
:<math>g_{m,n}(x) = e^{2\pi i m x} g(x - n),</math>
for integers ''m'' and ''n''. The Balian-Low theorem states that if
:<math>\{g_{m,n}: m, n \in \mathbb{Z}\}</math>
is an [[orthonormal basis]] for the [[Hilbert space]]
:<math>L^2(\mathbb{R}),</math>
then either
:<math> \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. </math>
The The Balian-Low theorem has been extended to exact Gabor [[Frame of a vector space|frames]].
== References ==
* {{cite journal
| author= John J. Benedetto, Christopher Heil, and David F. Walnut
| title= Differentiation and the Balian-Low Theorem
| year= 1994
| journal= Journal of Fourier Analysis and Applications
| volume= Volume 1, Number 4
| pages= 355–402
| doi= 10.1007/s00041-001-4016-5}}
== See also ==
* [[Gabor filter]] (in image processing)
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[[Category:Fourier analysis]]
[[Category:Mathematical theorems]]
[[fr:Théorème de Balian-Low]]