A survey on L2-approximation order from shift-invariant spaces

by Jetter, Kurt &

This paper aims at providing a self-contained introduction to notions and results connected with the $L_2$-approximation power of finitely generated shift-invariant spaces (FSI spaces) $S_{Phi} subset L_2(R^d)$. Here, approximation order refers to a scaling parameter and to the usual scaling of the $L_2$-projector onto $S_{Phi}$, where $Phi = {phi_1,dots,phi_n}subset L_2(R^d)$ is a given set of functions, the so-called generators of $S_{Phi}$. Special attention is given to the PSI case where the shift-invariant space is generated from the multi-integer translates of just one generator; this case is interesting enough due to 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 to this, the list of references is enlarged in order to give a rather comprehensive overview on existing literature in the field.