Parseval Frames Built Up from Generalized Shift-Invariant Systems

Abstract

Wavelet systems, and many of its generalizations such as wavelet packets, shearlets, and composite dilation wavelets are generalized shift-invariant systems (GSI) in the sense of the work by Ron and Shen. It is well known that a wavelet system is never Z-shift invariant (SI). Nevertheless, one can modify it and construct a Z-SI system, called a quasi-affine system, which shares most of the frame properties of the wavelet system. The analogue of a quasi-affine system for a GSI system is called an oblique oversampling: it is shift invariant with respect to a fixed lattice. Assumptions on a GSI system X were given by Ron and Shen to ensure that any oblique oversampling is a Parseval frame for ${L^2(\mathbb{R}^n)}$ whenever X is. We show that these assumptions are not satisfied for some of the wavelet generalizations mentioned above and that elements implicit in their work provide other sufficient conditions on the system under which any oblique oversampling is a Parseval frame for ${L^2(\mathbb{R}^n)}$ (shift invariant with respect to a fixed lattice). Moreover, in the orthonormal setting it is shown that completeness yields a shift-invariant Parseval frame for suitable proper subspaces of ${L^2(\mathbb{R}^n)}$, too.