Uniform convergence of Bernstein-Durrmeyer operators with respect to arbitrary measure

The Bernstein-Durrmeyer operator with respect to arbitrary measure is a modification of the classical Bernstein operator for functions from the corresponding weighted Lq-spaces on a simplex in Rd. As a first step in studying convergence of this operator, we consider uniform convergence. We prove that uniform convergence holds for all continuous functions if and only if the measure is strictly positive on the simplex. As a consequence, strict positivity of the measure is sufficient for convergence in the weighted Lq-spaces.