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).

Contents 1 Statement 2 Generalizations 3 See also 4 References Statement Suppose g is a square-integrable function on the real line, and consider the so-called Gabor system {displaystyle g_{m,n}(x)=e^{2pi imbx}g(x-na),} for integers m and n, and a,b>0 satisfying ab=1. The Balian–Low theorem states that if {displaystyle {g_{m,n}:m,nin mathbb {Z} }} is an orthonormal basis for the Hilbert space {displaystyle L^{2}(mathbb {R} ),} then either {displaystyle int _{-infty }^{infty }x^{2}|g(x)|^{2};dx=infty quad {textrm {or}}quad int _{-infty }^{infty }xi ^{2}|{hat {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, 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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Categories: Theorems in Fourier analysis