Abstract

This chapter aims at providing a self-contained introduction to notions and results connected with the L2-approximation order of finitely generated shift-invariant (FSI) spaces SΦ ⊂ L2(Rd). Here, the approximation order is with respect to a scaling parameter and to the usual scaling of the L2-projector onto SΦ, where Φ = {φ1, …, φn} ⊂ L2(Rd) is a given set of functions, the so-called generators of SΦ. Special attention is given to the principal shift-invariant (PSI) case, where the shift-invariant space is generated from the multi-integer translates of just one generator. This case is interesting in itself because of its possible applications in wavelet methods. The general FSI case is considered subject to a stability condition being satisfied, and the recent results on so-called superfunctions are developed. For the case of a refinable system of generators the sum rules for the matrix mask and the zero condition for the mask symbol, as well as invariance properties of the associated subdivision and transfer operator are discussed. References to the literature and further notes are extensively given at the end of each section. In addition, the list of references has been enlarged in order to provide a rather comprehensive overview of the existing literature in the field.

Introduction

In this chapter we give an overview on recent results concerning the L2-approximation order of so-called shift-invariant subspaces of L2(ℝd).