Iterates of $(\alpha,q)$-Bernstein operators

Abstract

In this paper, the iterates of $(\alpha,q)$-Bernstein operators are considered.
Given fixed $n\in{\mathbb N}$ and $q>0,$ it is shown that for $f\in C[0,1]$ the $k$-th iterate $T_{n,q,\alpha}^k(f;x)$ converges uniformly on $[0, 1]$ to the linear function $L_f(x)$ passing through the points $(0, f(0))$ and $(1,f(1)).$ Moreover, it is proved that, when $q\in(0,1),$ the iterates $T_{n,q,\alpha}^{j_n}(f;x),$ in which $\{j_n\}\to \infty$ as $n\to \infty,$ also converge to $L_f(x)$.
Further, when $q\in(1,\infty)$ and $\{j_n\}$ is a sequence of positive integers such that $j_n/[n]_q\to t$ as $n\to\infty,$ where $0\leq t\leq \infty,$ the convergence of the iterates $T_{n,q,\alpha}^{j_n}(p;x)$ for $p$ being a polynomial is studied.