Iterated Bernstein polynomial approximations

2009/09/03 by Zhong Guan, Zhong Zhen Guan, Guan, Zhong · 3 citations
Engineering · Mathematics · #41A10 #41A17 #41A25. #Advanced Numerical Analysis Techniques #Approximation Theory and Sequence Spaces #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Iterative Methods for Nonlinear Equations #math.CA #msc:41A10 #msc:41A17 #msc:41A25.

paper · pdf · doi:10.48550/arxiv.0909.0684

openalex publication_date 2009/09/03 · arxiv created 2009/10/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Iterated Bernstein polynomial approximations of degree n for continuous function which also use the values of the function at i/n, i=0,1,...,n, are proposed. The rate of convergence of the classic Bernstein polynomial approximations is significantly improved by the iterated Bernstein polynomial approximations without increasing the degree of the polynomials. The close form expression of the limiting iterated Bernstein polynomial approximation of degree n when the number of the iterations approaches infinity is obtained. The same idea applies to the q-Bernstein polynomials and the Szasz-Mirakyan approximation. The application to numerical integral approximations is also discussed.

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