On Shapiro's lethargy theorem and some applications

2012/11/14 by Asuman Güven Aksoy, A. G. Aksoy, Aksoy, A. G. +2 · 2 citations
Mathematics · #Mathematical Approximation and Integration #Mathematical Inequalities and Applications #Mathematical functions and polynomials #math.CA

paper · pdf · doi:10.48550/arxiv.1211.3356

22 pages, submitted to a Journal

arxiv created 2013/09/19 · arxiv updated 2013/09/20

Abstract

Shapiro's lethargy theorem states that if A_n is any non-trivial linear approximation scheme on a Banach space X, then the sequences of errors of best approximation E(x,A_n) = inf_{a ∈ An} ||x - a_n||_X decay almost arbitrarily slowly. Recently, Almira and Oikhberg investigated this kind of result for general approximation schemes in the quasi-Banach setting. In this paper, we consider the same question for F-spaces with non decreasing metric d. We also provide applications to the rate of decay of s-numbers, entropy numbers, and slow convergence of sequences of operators.

Cited by

• Bernstein's Lethargy Theorem in Frechet Spaces
• On a theorem of Nikol'skii

Related

• Shapiro's Theorem for subspaces
• Extensions of Bernstein's Lethargy Theorem
• Approximation schemes satisfying Shapiro's Theorem
• Rate of decay of s-numbers
• Characterization of approximation schemes satisfying Shapiro's Theorem