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).

StatementEdit

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

 

GeneralizationsEdit

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

See alsoEdit

ReferencesEdit

  • 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 10.1.1.118.7368. doi:10.1007/s00041-001-4016-5.

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