Abstract
Given a real functionf εC 2k[0,1],k ≥ 1 and the corresponding Bernstein polynomials {B n (f)} n we derive an asymptotic expansion formula forB n (f). Then, by applying well-known extrapolation algorithms, we obtain new sequences of polynomials which have a faster convergence thanB n (f). As a subclass of these sequences we recognize the linear combinations of Bernstein polynomials considered by Butzer, Frentiu, and May [2, 6, 9]. In addition we prove approximation theorems which extend previous results of Butzer and May. Finally we consider some applications to numerical differentiation and quadrature and we perform numerical experiments showing the effectiveness of the considered technique.
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This work was partially supported by a grant from MURST 40.
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Costabile, F., Gualtieri, M.I. & Serra, S. Asymptotic expansion and extrapolation for Bernstein polynomials with applications. Bit Numer Math 36, 676–687 (1996). https://doi.org/10.1007/BF01733787
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DOI: https://doi.org/10.1007/BF01733787