Approximation schemes satisfying Shapiro's Theorem

JOURNAL OF APPROXIMATION THEORY

卷:164

Article

Almira, J. M.¹  Oikhberg, T.^2,3 

[1] Univ Jaen, Dept Matemat, EPS Linares, Linares 23700, Jaen, Spain

[2] Univ Calif Irvine, Dept Math, Irvine, CA 92697 USA

[3] Univ Illinois, Dept Math, Urbana, IL 61801 USA

关键词: Approximation scheme;    Approximation error;    Bernstein's Lethargy;    Approximation by dictionary;   

DOI  :  10.1016/j.jat.2012.01.005

来源: Elsevier

【 摘 要 】

An approximation scheme is a family of homogeneous subsets (A(n)) of a quasi-Banach space X, such that A(1) subset of A(2) subset of ... subset of X, A(n) + A(n) subset of A(K(n)), and (U-n A(n)) over bar = X. Continuing the line of research originating at the classical paper [8] by Bernstein, we give several characterizations of the approximation schemes with the property that, for every sequence {epsilon(n)} SE arrow 0, there exists x is an element of X such that dist(x, A(n)) not equal O(epsilon(n)) (in this case we say that (X, {A(n)}) satisfies Shapiro's Theorem). If X is a Banach space, x is an element of X as above exists if and only if, for every sequence {delta(n)} SE arrow 0, there exists y is an element of X such that dist(y, A(n)) >= delta(n). We give numerous examples of approximation schemes satisfying Shapiro's Theorem. (C) 2012 Elsevier Inc. All rights reserved.

【 授权许可】

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