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

[math]\displaystyle{ g_{m,n}(x) = e^{2\pi i m b x} g(x - n a), }[/math]

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

[math]\displaystyle{ \{g_{m,n}: m, n \in \mathbb{Z}\} }[/math]

is an orthonormal basis for the Hilbert space

[math]\displaystyle{ L^2(\mathbb{R}), }[/math]

then either

[math]\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. }[/math]

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. doi:10.1007/s00041-001-4016-5.