Mathematics – Classical Analysis and ODEs
Scientific paper
2010-06-02
Mathematics
Classical Analysis and ODEs
4 pages, Submitted to a Journal
Scientific paper
In this short note we prove that, if (C[a,b],{A_n}) is an approximation scheme and (A_n) satisfies de La Vall\'ee-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).
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