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
- Gabor filter (in image processing)