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

摘要

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 p 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.