From exact systems to Riesz bases in the Balian-Low theorem

TL;DR

This paper extends the classical Balian-Low theorem to more general Gabor systems, establishing limits on time-frequency localization based on approximation properties and covering both symmetric and non-symmetric cases.

Contribution

It introduces Balian-Low type theorems for complete and minimal Gabor systems with frame approximation, bridging classical and generalized conditions.

Findings

Limits on localization depend on approximation control

Results apply to symmetric and non-symmetric systems

Provides a continuous transition between classical and generalized conditions

Abstract

We look at the time-frequency localisation of generators of lattice Gabor systems. For a generator of a Riesz basis, this localisation is described by the classical Balian-Low theorem. We establish Balian-Low type theorems for complete and minimal Gabor systems with a frame-type approximation property. These results describe how the best possible localisation of a generator is limited by the degree of control over the coefficients in approximations given by the system, and provide a continuous transition between the classical Balian-Low conditions and the corresponding conditions for generators of complete and minimal systems. Moreover, this holds for the non-symmetric generalisations of these theorems as well.