Iterated Bernstein polynomial approximations

TL;DR

This paper introduces iterated Bernstein polynomial approximations that enhance convergence rates for continuous functions without increasing polynomial degree, extending the approach to q-Bernstein and Szasz-Mirakyan approximations, with applications to numerical integration.

Contribution

It proposes a novel iterated Bernstein polynomial method that improves approximation convergence and derives its limiting form, extending the concept to related polynomial approximations.

Findings

Convergence rate of Bernstein approximations is significantly improved.

Derived the limiting form of the iterated Bernstein polynomial as iterations approach infinity.

Extended the approach to q-Bernstein and Szasz-Mirakyan approximations.

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.