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

Abstract (Summary)

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.
Bibliographical Information:

Advisor:none

School:Universität Duisburg-Essen, Standort Essen

School Location:Germany

Source Type:Master's Thesis

Keywords:mathematik gerhard mercator universitaet

ISBN:

Date of Publication:05/27/2002