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).
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
The Balian–Low theorem has been extended to exact Gabor frames.
See also
- Gabor filter (in image processing)
References
- Benedetto, John J.; Heil, Christopher; Walnut, David F. (1994). "Differentiation and the Balian–Low Theorem". Journal of Fourier Analysis and Applications 1 (4): 355–402. doi:10.1007/s00041-001-4016-5.
This article incorporates material from Balian-Low on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
This article is issued from Wikipedia - version of the 7/11/2014. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.