An iterative algorithm for improved approximation by Bernstein’s operator using statistical perspective

Abstract

The celebrated Weierstrass Approximation Theorem (1885) heralded intermittent interest in polynomial approximation, which continues unabated even as of today. The great Russian mathematician Bernstein in 1912 not only provided an interesting proof of the Weierstrass’ theorem, but also displayed a sequence of polynomials that approximate a given function f∈ C[0,1]. This paper dwells upon constructing an iterative algorithm with an intention of improving the degree of approximation by this Bernstein’s polynomial. The perspective motivating the proposed algorithm is Statistical, and works through a fuller use of the information, about the unknown function f being approximated, which is available in terms of its known values at equidistant knots in C[0,1]. This information is used aposteriori to improve upon the approximation by the Bernstein polynomial. This improvement turns out to be available iteratively, which is seminal to the iterative nature of the proposed algorithm of the improvement of the degree of approximation by the Bernstein polynomial. The potential of the aforesaid improvement algorithm is tried to be brought forth and illustrated through an empirical study for which the function is assumed to be known in the sense of simulation.