On the rates of pointwise convergence for Bernstein polynomials

TL;DR

This paper investigates the pointwise convergence rates of Bernstein polynomials for functions with constant segments, showing exponential decay inside the interval and boundary limitations, with extensions to Bernstein-Kantorovich operators.

Contribution

It establishes exponential convergence rates for Bernstein polynomials on constant segments and explores boundary behavior, extending results to Bernstein-Kantorovich operators.

Findings

Exponential convergence of Bernstein polynomials inside the interval.

Boundary points do not exhibit the same convergence rate.

Extension of results to Bernstein-Kantorovich operators.

Abstract

Let $f$ be a real function defined on the interval $[0,1]$ which is constant on $(a,b)\subset [0,1]$, and let $B_nf$ be its associated $n$th Bernstein polynomial. We prove that, for any $x\in (a,b)$, $|B_nf(x)-f(x)|$ converges to $0$ as $n\rightarrow \infty $ at an exponential rate of decay. Moreover, we show that this property is no longer true at the boundary of $(a,b)$. Finally, an extension to Bernstein-Kantorovich type operators is also provided