Bernstein–Durrmeyer operators with respect to arbitrary measure, II: Pointwise convergence

We consider the Bernstein–Durrmeyer operator M n , ρ with respect to an arbitrary measure ρ on the d-dimensional simplex. This operator is a generalization of the well-known Bernstein–Durrmeyer operator with respect to the Lebesgue measure. We prove that ( M n , ρ f ) ( x ) → f ( x ) as n → ∞ at each point x ∈ supp ρ if f is bounded on supp ρ and continuous at x. Moreover, the convergence is uniform in any compact set in the interior of supp ρ.