A Balian-Low Theorem for Subspaces

TL;DR

This paper extends the Balian-Low theorem to Gabor subspaces, showing that if such a subspace is invariant under extra time-frequency shifts, its generator cannot decay rapidly in both time and frequency.

Contribution

It introduces a new extension of the Balian-Low theorem involving additional invariance under time-frequency shifts for Gabor subspaces.

Findings

Gabor systems with rational lattice density cannot have generators with rapid decay if invariant under extra shifts.

The theorem applies to Riesz sequences generating invariant subspaces.

The result links invariance properties to decay constraints in time-frequency analysis.

Abstract

We extend the Balian-Low theorem to Gabor subspaces of $L^2(\mathbb R)$ by involving the concept of additional time-frequency shift invariance. We prove that if a Gabor system on a lattice of rational density is a Riesz sequence generating a subspace which is invariant under an additional time-frequency shift, then its generator cannot decay fast simultaneously in time and frequency.