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

In this paper we introduce and study a sequence of Bernstein-Durrmeyer type operators $(M_{n,mu})_{ngeq 1}$, acting on spaces of continuous or integrable functions on the multi-dimensional hypercube $Q_d$ of $mathbf{R}^d$ ($dgeq 1$), 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.