Shift-invariant spline subspaces of Sobolev spaces and their order of approximation

Abstract

Closed shift-invariant subspaces of Hilbert space are the core idea of important classes of signal and image analysis methods such as wavelets, shearlets, radial basis functions and many more. In 1994 de Boor, DeVore and Ron [4] gave a full characterization of the order of approximation in such shift-invariant subspaces of L2(Rn), n∈N. Ron and Holtz extended this work in [10] to Sobolev norms, and gave specific criteria on the generating functions to guarantee certain orders of approximation. However, not for all generating functions these criteria are easy to verify. We here give new criteria which are designed for generating functions given explicitly in Fourier domain. We show by various examples on spline functions that in these cases our new criteria give better estimates on the order of approximation.