Korovkin-type approximation theory and its applications
type: Item
instance of: scholarly article
title: Korovkin-type approximation theory and its applications
zbMATH Open document ID: 0924.41001
author: Francesco Altomare
author: Michele Campiti
published in: De Gruyter Studies in Mathematics
publication date: 1994-12-11T00:00:00Z
review text: In 1953 P. P. Korovkin discovered a criterion in order to decide whether a given sequence \((L_n)\) of positive linear operators on the space \(C[0,1]\) is an approximation process, i.e., \(L_nf\to f\) uniformly on [0,1] for every \(f\in C[0,1]\): in fact, it is sufficient to verify that \(L_nf\to f\) only for \(f\in \{1,x,x^2\}\). This result has been extended to other function spaces and to abstract spaces. Now the Korovkin approximation theory (KAT) has fruitful connections with functional analysis (notably with Choquet's theory and Banach algebras theory), harmonic analysis, measure and probability theory, partial differential equations; an important feature of this monograph is the systematic presentation of all these connections. The book is addressed to specialists in the above mentioned fields; a large part of it can also serve as a textbook for a graduate-level course. The authors have contributed significantly to the development of this theory. They give a survey on classical as well as recent developments in the field. Most of the results appear in book form for the first time. In order to make the exposition self-contained, the prerequisites (from topology and analysis, measure and probability theory, locally convex spaces and Choquet's integral representation theory, semigroups of operators) are collected in Chapter 1. Chapters 2, 3 and 4 are devoted to the main aspects of KAT in \(C_0(X)\) \((X\) locally compact) and \(C(X)\) \((X\) compact). The fundamental problem consists in studying, for a given positive linear operator \(T:C_0(X)\to C_0(Y)\), those subspaces \(H\) of \(C_0(X)\) (if any) which have the property that every equicontinuous net of positive linear operators (or positive contractions) from \(C_0(X)\) into \(C_0(Y)\) converges strongly to \(T\) whenever it converges to \(T\) on \(H\). (Such subspaces are called Korovkin subspaces for \(T\)). The classical case when \(T\) is the identity operator is discussed in Chapter 4, where the author…
Mathematics Subject Classification ID: 41-02
Mathematics Subject Classification ID: 41A36
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 41A35
Mathematics Subject Classification ID: 41A63
zbMATH DE Number: 700798
zbMATH Keywords: Korovkin approximation theory
reviewed by: Ioan Raşa
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 700798