A Sharp Balian-Low Uncertainty Principle for Shift-Invariant Spaces

A sharp version of the Balian-Low theorem is proven for the generators of finitely generated shift-invariant spaces. If generators $\{f_k\}_{k=1}^K \subset L^2(\mathbb{R}^d)$ are translated along a lattice to form a frame or Riesz basis for a shift-invariant space $V$, and if $V$ has extra invariance by a suitable finer lattice, then one of the generators $f_k$ must satisfy $\int_{\mathbb{R}^d} |x| |f_k(x)|^2 dx = \infty$, namely, $\widehat{f_k} \notin H^{1/2}(\mathbb{R}^d)$. Similar results are proven for frames of translates that are not Riesz bases without the assumption of extra lattice invariance. The best previously existing results in the literature give a notably weaker conclusion using the Sobolev space $H^{d/2+\epsilon}(\mathbb{R}^d)$; our results provide an absolutely sharp improvement with $H^{1/2}(\mathbb{R}^d)$. Our results are sharp in the sense that $H^{1/2}(\mathbb{R}^d)$ cannot be replaced by $H^s(\mathbb{R}^d)$ for any $s<1/2$.