A lethargy result for real analytic functions

2010/06/02 by J. M. Almira, Almira, J. M.
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA

paper · pdf · doi:10.48550/arxiv.1006.0413

4 pages, Submitted to a Journal

arxiv created 2011/11/11 · arxiv updated 2011/11/14

Abstract

In this short note we prove that, if (C[a,b],A_n) is an approximation scheme and (A_n) satisfies de La Vallée-Poussin Theorem, there are instances of continuous functions on [a,b], real analytic on (a,b], which are poorly approximable by the elements of the approximation scheme (A_n). This illustrates the thesis that the smoothness conditions guaranteeing that a function is well approximable must be, at least in these cases, global. The failure of smoothness at endpoints may result in an arbitrarily slow rate of approximation. A result of this kind, which is highly nonconstructive, based on different arguments, and applicable to different approximation schemes, was recently proved by Almira and Oikhberg (see arXiv:1009.5535v2).

Citations

• Shapiro's Theorem for subspaces

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