Asymptotic expansion and extrapolation for Bernstein polynomials with applications (Q2565270)

This paper generalizes results of \textit{P. L. Butzer} [Canadian J. Math. 5, 559-567 (1953; Zbl 0051.05002)] \textit{M. Frentiu} [Studia Univ. Babes-Bolýai, Ser. Math. Mech. 15, No. 1, 63-68 (1970; Zbl 0283.41002)] and \textit{C. P. May} [Canadian J. Math. 28, 1224-1250 (1976; Zbl 0342.41018)] for approximation using sequences of Bernstein polynomials. The basic idea is to show that the \(n+1\) point Bernstein polynomial approximation, to a smooth function \(f(x)\) on \([0,1]\), has an error expansion in increasing powers of the mesh size \(h= 1/n\). Sequences of these approximations can then be extrapolated, using Richardson extrapolation, to produce successively more accurate approximations to \(f(x)\). The authors consider the most general case, where the mesh sequence depends only on an increasing sequence of \(n\) values. The authors prove a general error bound for this case. They conclude with a discussion of how the extrapolated polynomials can be used for quadrature. The paper finishes with some numerical examples.