Multivariate Bernstein–Durrmeyer operators with arbitrary weight functions

In this paper we introduce a class of Bernstein–Durrmeyer operators with respect to an arbitrary measure ρρ on the dd-dimensional simplex, and a class of more general polynomial integral operators with a kernel function involving the Bernstein basis polynomials. These operators generalize the well-known Bernstein–Durrmeyer operators with respect to Jacobi weights. We investigate properties of the new operators. In particular, we study the associated reproducing kernel Hilbert space and show that the Bernstein basis functions are orthogonal in the corresponding inner product. We discuss spectral properties of the operators. We make first steps in understanding convergence of the operators.