A Balian–Low Theorem for Subspaces

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.