Abstract
We consider the multivariate Bernstein–Durrmeyer operator Mn,μ in terms of the Choquet integral with respect to a monotone and submodular set function μ on the standard d-dimensional simplex. This operator is nonlinear and generalizes the Bernstein–Durrmeyer linear operator with respect to a nonnegative, bounded Borel measure (including the Lebesgue measure). We prove uniform and pointwise convergence of Mn,μ(f)(x) to f(x) as n→∞, generalizing thus the results obtained in the recent papers [1] and [2].
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