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

We consider the Bernstein–Durrmeyer operator Mn,ρMn,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 (Mn,ρf)(x)→f(x)(Mn,ρf)(x)→f(x) as n→∞n→∞ at each point x∈suppρ 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.