Determination of Chebyshev approximations by nonlinear admissible subsets
type: Item
instance of: scholarly article
title: Determination of Chebyshev approximations by nonlinear admissible subsets
zbMATH Open document ID: 0532.41038
DOI: 10.1016/0021-9045(83)90117-X
author: Ryszard Smarzewski
published in: Journal of Approximation Theory
publication date: 1983-01-01T00:00:00Z
review text: Let B(X) denote the linear space of all bounded real-valued functions defined on a set X. In particular let \(w\in B(X)\) be such that \(\delta =\inf \{w(x): x\in X\}>0\). A weighted uniform norm \(\| \cdot \|_ w\) on B(X) is defined by \(\| f\|_ w=sp\{w(x)| f(x)|:\) \(x\in X\}\). Let G be a nonempty proper subset of B(X) and let f be a fixed element of B(X)\(\backslash G\). Denote \(\theta_ f=\inf \{\| f- h\|_ w:\quad h\in G\}.\) An element \(g\in G\) such that \(\theta_ f=\| f-g\|_ w\) is called the best weighted approximation to f in G. Let \(G_ f\) be the set of all such elements in G. Using the usual partial ordering of B(X) (\(f\leq g\) meaning f(x)\(\leq g(x)\) for all \(x\in X)\), the author specifies the notion of an ''admissible'' subset G of B(X) with respect to the pair \((f_ 1,f_ 2)\in B^ 2(X)\) and determines the set \(G_ f\) of all best weighted approximations to f by elements of an admissible subset G with respect to the pair \((f-\theta_ f/w,f+\theta_ f/w)\). In the sequel he applies his general theory if G is in particular: the set of semi-isotone functions, the set of functions whose modulus of continuity is bounded by a given modulus of continuity, the set of even functions. Some of these results develops the recent results of \textit{V. A. Ubhaya} [J. Approximation Theory 12, 146-159 (1974; Zbl 0288.41019), and ibid. 12, 315-331 (1974; Zbl 0292.41026); J. Math. Anal. Appl. 49, 659-679 (1975; Zbl 0299.41025)].
reviewed by: S.Aljančić
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 41A30
Mathematics Subject Classification ID: 41A17
zbMATH DE Number: 3844332
zbMATH Keywords: best weighted approximation
zbMATH Keywords: semi-isotone functions
zbMATH Keywords: even functions
MaRDI profile type: Publication
cites work: Q5331549
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cites work: Some remarks on nonlinear Chebyshev approximation to functions defined on normal spaces
cites work: Isotone optimization. I
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description: scientific article; zbMATH DE number 3844332