Английская Википедия:Balian–Low theorem

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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>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>\{g_{m,n}: m, n \in \mathbb{Z}\}</math>

is an orthonormal basis for the Hilbert space

<math>L^2(\mathbb{R}),</math>

then either

<math> \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

• Шаблон:Cite journal