Iterates of (α,q)−Bernstein operators

Abstract

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