Polynomial approximations to continuous functions and stochastic compositions



Konstantopoulos, Takis, Yuan, Linglong and Zazanis, Michael A
(2016) Polynomial approximations to continuous functions and stochastic compositions. Unknown.

[thumbnail of 1601.04483v1.pdf] Text
1601.04483v1.pdf - Submitted version

Download (294kB)
[thumbnail of 1601.04483v1.pdf] Text
1601.04483v1.pdf - Submitted version

Download (294kB)

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.

Item Type: Article
Additional Information: 21 pages, 5 figures
Uncontrolled Keywords: math.PR, math.PR, 60J10, 41A10, 41-01 (Primary), 60H30 (Secondary)
Depositing User: Symplectic Admin
Date Deposited: 22 Aug 2018 14:32
Last Modified: 19 Jan 2023 01:26
Related URLs:
URI: https://livrepository.liverpool.ac.uk/id/eprint/3025410