A lethargy result for real analytic functions
Copyright policy )A lethargy result for real analytic functions
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).
4 pages, Submitted to a Journal
- University of Jaén Spain
Approximation scheme, approximation error, Applied Mathematics, approximation scheme, De La Vallée poussin theorem, Smoothness, Approximation error, Best approximation, Chebyshev systems, Abstract approximation theory (approximation in normed linear spaces and other abstract spaces), Haar space, Mathematics - Classical Analysis and ODEs, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Bernstein’s lethargy, Bernstein's lethargy, smoothness, De La Vallée Poussin theorem
Approximation scheme, approximation error, Applied Mathematics, approximation scheme, De La Vallée poussin theorem, Smoothness, Approximation error, Best approximation, Chebyshev systems, Abstract approximation theory (approximation in normed linear spaces and other abstract spaces), Haar space, Mathematics - Classical Analysis and ODEs, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Bernstein’s lethargy, Bernstein's lethargy, smoothness, De La Vallée Poussin theorem
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