Density order of Parseval wavelet frames from extension principles

Abstract

We characterize approximation order and density order of those Parseval wavelet frames obtained from Oblique Extension Principle. These notions are closely related to approximation order and density order by a quasi-projection operator. To give our characterizations, we shall explain the behavior on a neighborhood of the origin of the Fourier transform of a refinable function. In particular, we invoke the classical notion of approximate continuity. We write our results in the multivariate context of Parseval wavelet frames associated to A, an expansive linear map preserving the integer lattice.