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, t...

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Bibliographic Details
Published in:Applied and computational harmonic analysis 2018-03, Vol.44 (2), p.294-311
Main Authors: Hardin, Douglas P., Northington, Michael C., Powell, Alexander M.
Format: Article
Language:English
Subjects:
Balian–Low theorem
Shift-invariant spaces
Uncertainty principle
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Summary: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
ISSN:1063-5203
1096-603X
DOI:10.1016/j.acha.2016.05.001