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A survey on \(L_2\)-approximation orders from shift-invariant spaces. (English) Zbl 1039.42033

Dyn, N. (ed.) et al., Multivariate approximation and applications. Cambridge: Cambridge University Press (ISBN 0-521-80023-4/hbk). 73-111 (2001).
Summary: This chapter aims at providing a self-contained introduction to notions and results connected with the \(L_2\)-approximation order of finitely generated shift-invariant (FSI) spaces \(S_\Phi \subset L_2(\mathbb{R}^d)\). Here, the approximation order is with respect to a scaling parameter and to the usual scaling of the \(L_2\)-projector onto \(S_\Phi\), where \(\Phi= \{\varphi_1,\dots, \varphi_n\} \subset L_2(\mathbb{R}^d)\) is a given set of functions, the so-called generators of \(S_\Phi\). 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.
For the entire collection see [Zbl 0963.00017].

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
41A30 Approximation by other special function classes
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
41A25 Rate of convergence, degree of approximation
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