Balian–Low theorem

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 i m b x} g(x - n a),

for integers m and n, and a,b0 satisfying ab=1. The Balian–Low theorem states that if

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

is an orthonormal basis for the Hilbert space

:L^2(\mathbb{R}),

then either : \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.

References

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