Balian-Low theorem

In mathematics, the Balian-Low theorem in Fourier analysis is named for Roger Balian and Francis E. Low.

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

:$g\_\{m,n\}(x)\; =\; e^\{2pi\; i\; m\; x\}\; g(x\; -\; n),$

for integers "m" and "n". The Balian-Low theorem states that if

:$\{g\_\{m,n\}:\; m,\; n\; in\; mathbb\{Z\}\}$

is an orthonormal basis for the Hilbert space

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

then either:$int\_\{-infty\}^infty\; x^2\; |\; g(x)|^2;\; dx\; =\; infty\; quad\; extrm\{or\}\; quad\; int\_\{-infty\}^infty\; xi^2|hat\{g\}(xi)|^2;\; dxi\; =\; infty.$

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

References

* cite journal
author= John J. Benedetto, Christopher Heil, and David F. Walnut
title= Differentiation and the Balian-Low Theorem
year= 1994
journal= Journal of Fourier Analysis and Applications
volume= Volume 1, Number 4
pages= 355–402
doi= 10.1007/s00041-001-4016-5

See also

* Gabor filter (in image processing)

----