A lethargy result for real analytic functions

TL;DR

This paper demonstrates that real analytic functions can be poorly approximable by certain schemes, highlighting that local smoothness at endpoints does not guarantee good approximation, emphasizing the importance of global smoothness conditions.

Contribution

It proves the existence of continuous, real analytic functions that are poorly approximable by schemes satisfying de La Vallée-Poussin Theorem, extending understanding of approximation limitations.

Findings

Existence of poorly approximable real analytic functions.

Smoothness at endpoints does not ensure good approximation.

Global smoothness conditions are necessary for effective approximation.

Abstract

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).