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

${\displaystyle g_{m,n}(x)=e^{2\pi imbx}g(x-na),}$ 

for integers m and n, and a,b>0 satisfying ab=1. The Balian–Low theorem states that if

${\displaystyle \{g_{m,n}:m,n\in \mathbb {Z} \}}$ 

is an orthonormal basis for the Hilbert space

${\displaystyle L^{2}(\mathbb {R} ),}$ 

then either

${\displaystyle \int _{-\infty }^{\infty }x^{2}|g(x)|^{2}\;dx=\infty \quad {\textrm {or}}\quad \int _{-\infty }^{\infty }\xi ^{2}|{\hat {g}}(\xi )|^{2}\;d\xi =\infty .}$ 

Generalizations

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. CiteSeerX 10.1.1.118.7368. 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.