Approximation by polynomials with locally geometric rates

Ivanov, K. G.; Saff, E. B.; Totik, V.

doi:10.1090/S0002-9939-1989-0964456-3

Approximation by polynomials with locally geometric rates
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by K. G. Ivanov, E. B. Saff and V. Totik PDF

Proc. Amer. Math. Soc. 106 (1989), 153-161 Request permission

Abstract:

In contrast to the behavior of best uniform polynomial approximants on $[0,1]$ we show that if $f \in C[0,1]$ there exists a sequence of polynomials $\{ {P_n}\}$ of respective degree $\leq n$ which converges uniformly to $f$ on $[0,1]$ and geometrically fast at each point of $[0,1]$ where $f$ is analytic. Moreover we describe the best possible rates of convergence at all regular points for such a sequence.

References

K. G. Ivanov and V. Totik, Fast decreasing polynomials, Constr. Approx. 6 (1990), no. 1, 1–20. MR 1027506, DOI 10.1007/BF01891406
M. Ĭ. Kadec′, On the distribution of points of maximum deviation in the approximation of continuous functions by polynomials, Amer. Math. Soc. Transl. (2) 26 (1963), 231–234. MR 0151766, DOI 10.1090/trans2/026/09
E. B. Saff and V. Totik, Polynomial approximation of piecewise analytic functions, J. London Math. Soc. (2) 39 (1989), no. 3, 487–498. MR 1002461, DOI 10.1112/jlms/s2-39.3.487
A. F. Timan, Theory of approximation of functions of a real variable, A Pergamon Press Book, The Macmillan Company, New York, 1963. Translated from the Russian by J. Berry; English translation edited and editorial preface by J. Cossar. MR 0192238