Oblique dual frames and shift-invariant spaces

Abstract

Given a frame for a subspace W of a Hilbert space H , we consider a class of oblique dual frame sequences. These dual frame sequences are not constrained to lie in W . Our main focus is on shift-invariant frame sequences of the form {φ(·−k)} k∈ Z in subspaces of L 2( R) ; for such frame sequences we are able to characterize the set of shift-invariant oblique dual Bessel sequences. Given frame sequences {φ(·−k)} k∈ Z and {φ 1(·−k)} k∈ Z , we present an easily verifiable condition implying that span {φ 1(·−k)} k∈ Z contains a generator for a shift-invariant dual of {φ(·−k)} k∈ Z ; in particular, the exact statement of this result implies the somewhat surprising fact that there is a unique conventional dual frame that is shift-invariant. As an application of our results we consider frame sequences generated by B-splines, and show how to construct oblique duals with prescribed regularity.