On Shapiro's lethargy theorem and some applications

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 \in A_n} ||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.

22 pages, submitted to a Journal

Cited by in corpus (4)

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• Bernstein Lethargy Theorem and Reflexivity
• Subspace Condition for Bernstein's Lethargy Theorem