Balian–Low theorem

Remarks

So a window of a Gabor Riesz basis at the critical density $ab = 1$ cannot be well localized both in time and in frequency. Roger Balian and, independently, Francis Low stated the result for orthonormal bases; later work completed their proofs and extended it to Riesz bases. A standard route uses the Zak transform: the system is a Riesz basis exactly when $Zg$ is bounded above and below by positive constants almost everywhere. On the other hand $Zg$ is quasi-periodic, and a continuous quasi-periodic function must vanish somewhere; when $g$ is well localized this contradicts the lower bound. The hypothesis $ab = 1$ matters: for $ab < 1$ the Gaussian $e^{-\pi t^2}$, which is as well localized as possible, generates a Gabor frame.

Example: $g = 1_{[0, 1)}$ generates an orthonormal basis. Here $\int \abs{t g(t)}^2 \, dt = \tfrac13$ is finite, but $\abs{\hat g(\xi)} = \abs{\sin(\pi \xi)}/(\pi \abs{\xi})$, so $\abs{\xi \hat g(\xi)}^2 = \sin^2(\pi \xi)/\pi^2$ is not integrable.

Going deeper

For a student

The theorem explains why practical Gabor expansions are redundant frames with $ab < 1$ rather than bases. It is proved with the Zak transform, which turns the Gabor system at $a = b = 1$ into multiplication by a quasi-periodic function, and the question becomes one about zeros of that function. It contrasts with wavelets: there exist orthonormal wavelet bases, such as the Meyer wavelets, whose generator is smooth and decays fast together with its Fourier transform. Compare with the uncertainty principle in Fourier analysis, which bounds the product of the two integrals from below for one function.