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 {fk}k=1K⊂L2(Rd) 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 fk must satisfy ∫Rd|x||fk(x)|2dx=∞, namely, fkˆ∉H1/2(Rd). 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 Hd/2+ϵ(Rd); our results provide an absolutely sharp improvement with H1/2(Rd). Our results are sharp in the sense that H1/2(Rd) cannot be replaced by Hs(Rd) for any s<1/2.