Balian-Low theorem 4795347 215568906 2008-05-28T19:53:21Z DOI bot 6652755 Citation maintenance. You can [[WP:DOI|use this bot]] yourself! Please [[User:DOI_bot/bugs|report any bugs]]. 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) ---- {{mathanalysis-stub}} {{planetmath|id=7497|title=Balian-Low}} [[Category:Fourier analysis]] [[Category:Mathematical theorems]] [[fr:Théorème de Balian-Low]]