Abstract

We consider the Bernstein–Durrmeyer operator Mn,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) as n→∞ at each point x∈suppρ if f is bounded on suppρ and continuous at x. Moreover, the convergence is uniform in any compact set in the interior of suppρ.

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