A generalization of Bernstein-Durrmeyer operators on hypercubes by means of an arbitrary measure

Abstract

In this paper we introduce and study a sequence of Bernstein-Durrmeyer type operators  $(M_{n,\mu})_{n\geq 1}$,  acting on spaces of continuous or integrable functions on the multi-dimensional hypercube $Q_d$ of $\mathbf{R}^d$ ($d\geq 1$), defined by means of an arbitrary measure  $\mu$. We investigate their approximation properties both in the space of all continuous functions and in $L^p$-spaces with respect to $\mu$, also furnishing some esitmates of the rate of convergence. Further, we prove  an asymptotic formula for the $M_{n,\mu}$'s. The paper ends with a concrete example.