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 g either in time or frequency for an exact Gabor frame.

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.