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).

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

g_{{m,n}}(x)=e^{{2\pi imbx}}g(x-na),

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

\{g_{{m,n}}:m,n\in {\mathbb {Z}}\}

L^{2}({\mathbb {R}}),

then either

\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 .

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

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

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