Oblique dual frames and shift-invariant spaces

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 {phi((.) - k)}(kis an element ofZ) 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 {phi((.) - k)}(kis an element ofZ) and {phi(1)((.) - k)}(kis an element ofZ), we present an easily verifiable condition implying that span{phi(1)((.) - k)}(kis an element ofZ) contains a generator for a shift-invariant dual of {phi((.) - k)}(kis an element ofZ); 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. (C) 2004 Published by Elsevier Inc.