Oblique dual frames and shift-invariant spaces

AbstractGiven 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 L2(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 {φ1(·−k)}k∈Z, we present an easily verifiable condition implying that span{φ1(·−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