Convergence in distribution of the Bernstein-Durrmeyer kernel and pointwise convergence of a generalised operator for functions of bounded variation

Abstract

We study the convergence of Bernstein type operators leading to two results. The first: The kernel $K_n$ of the Bernstein-Durrmeyer operator at each point $x \in (0, 1)$ $\unicode{x2013}$ that is $K_n(x, t) dt$ $\unicode{x2013}$ once standardised converges to the normal distribution. The second result computes the pointwise limit of a generalised Bernstein-Durrmeyer operator applied to $\unicode{x2013}$ possibly discontinuous $\unicode{x2013}$ functions $f$ of bounded variation.