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

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

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

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

is an orthonormal basis for the Hilbert space

$L^2(\mathbb{ℝ}),$

then either

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{x}^2|g(x){|}^{2}dx=\mathrm{\infty }\text{<mrow data-mjx-texclass="ORD">or</mrow>}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{\xi }^2|\hat{g}(\xi ){|}^{2}d\xi =\mathrm{\infty }.$

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