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).

Statement

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

Generalizations

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

See also