# Balian–Low theorem

Balian–Low theorem 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).

Inhalt 1 Aussage 2 Verallgemeinerungen 3 Siehe auch 4 References Statement Suppose g is a square-integrable function on the real line, and consider the so-called Gabor system {Anzeigestil g_{m,n}(x)=e^{2pi imbx}g(x-na),} for integers m and n, und ein,b>0 satisfying ab=1. The Balian–Low theorem states that if {Anzeigestil {g_{m,n}:m,nin mathbb {Z} }} is an orthonormal basis for the Hilbert space {Anzeigestil L^{2}(mathbb {R} ),} dann entweder {Anzeigestil int _{-unendlich }^{unendlich }x^{2}|g(x)|^{2};dx=infty quad {textrm {oder}}quad int _{-unendlich }^{unendlich }xi ^{2}|{Hut {g}}(xi )|^{2};dxi =infty .} Generalizations The Balian–Low theorem has been extended to exact Gabor frames.

See also Gabor filter (in image processing) References Benedetto, John J.; Heil, Christoph; 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.