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. The theorem states that there is no well-localized window function (or Gabor atom) g either in time or frequency for an exact Gabor frame (Riesz Basis).


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


for integers m and n, and a,b>0 satisfying ab=1. The Balian–Low theorem states that if


is an orthonormal basis for the Hilbert space


then either



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

See alsoEdit


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

This article incorporates material from Balian-Low on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.