A sharp Balian-Low uncertainty principle for shift-invariant spaces

作者：Hardin Douglas P; Northington Michael C V; Powell Alexander M^*

来源：Applied and Computational Harmonic Analysis, 2018, 44(2): 294-311.

DOI：10.1016/j.acha.2016.05.001

• Balian-Low theorem
• Shift-invariant spaces
• Uncertainty principle

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) C=subset of L-2(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 integral(Rd) vertical bar x vertical bar vertical bar f(k)(x)vertical bar(2)d(x) = infinity, namely, (f(k)) over cap is not an element of H-1/2(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 Hd/2+is an element of(R-d); our results provide an absolutely sharp improvement with H-1/2(R-d). Our results are sharp in the sense that H-1/2(R-d) cannot be replaced by H-s(R-d) for any s < 1/2.

• 出版日期2018-3