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 ${\int }_{{\mathbb{R}}^d}\mathrm{\mid }x\mathrm{\mid }\mathrm{\mid }{f}_k(x){\mathrm{\mid }}^{2}dx=\mathrm{\infty }$, 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 $H^{d\mathrm{/}2+ϵ}({\mathbb{R}}^d)$; our results provide an absolutely sharp improvement with $H^{1\mathrm{/}2}({\mathbb{R}}^d)$. Our results are sharp in the sense that $H^{1\mathrm{/}2}({\mathbb{R}}^d)$ cannot be replaced by $H^s({\mathbb{R}}^d)$ for any $s<1\mathrm{/}2$.