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)
References