Polynomial approximations to continuous functions and stochastic compositions

TL;DR

This paper investigates the behavior of polynomial approximations to continuous functions using stochastic processes like Wright-Fisher models, providing probabilistic explanations and formulas for iterative Bernstein operators.

Contribution

It introduces a stochastic approach to analyze Bernstein operator iterations, connecting them with Wright-Fisher models and deriving new probabilistic formulas.

Findings

Probabilistic explanation of Kelisky-Rivlin theorem

Formulas for iterated Bernstein operators with growing iterations

Connection between polynomial approximation and stochastic processes

Abstract

This paper presents a stochastic approach to theorems concerning the behavior of iterations of the Bernstein operator $B_n$ taking a continuous function $f \in C[0,1]$ to a degree-$n$ polynomial when the number of iterations $k$ tends to infinity and $n$ is kept fixed or when $n$ tends to infinity as well. In the first instance, the underlying stochastic process is the so-called Wright-Fisher model, whereas, in the second instance, the underlying stochastic process is the Wright-Fisher diffusion. Both processes are probably the most basic ones in mathematical genetics. By using Markov chain theory and stochastic compositions, we explain probabilistically a theorem due to Kelisky and Rivlin, and by using stochastic calculus we compute a formula for the application of $B_n$ a number of times $k=k(n)$ to a polynomial $f$ when $k(n)/n$ tends to a constant.