Uniform and pointwise convergence of Bernstein–Durrmeyer operators with respect to monotone and submodular set functions

We consider the multivariate Bernstein–Durrmeyer operator Mn,μMn,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)Mn,μ(f)(x) to f(x)f(x) as n→∞n→∞, generalizing thus the results obtained in the recent papers [1] and [2].