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For the qualitative results of pointwise and uniform approximation obtainedin \cite{Gal-Opris}, we present general quantitative estimates in terms of themodulus of continuity and in terms of a $K$-functional, for the generalizedmultivariate Bernstein-Durrmeyer operator $M_{n, \Gamma_{n, x}}$, written interms of the Choquet integral with respect to a family of monotone andsubmodular set functions, $\Gamma_{n, x}$, on the standard $d$-dimensionalsimplex. When $\Gamma_{n, x}$ reduces to two elements, one a Choquet submodularset function and the other one a Borel measure, for suitable modifiedBernstein-Durrmeyer operators, univariate $L^{p}$-approximations, $p\ge 1$,with estimates in terms of a $K$-functional are proved. In the particular caseswhen $d=1$ and the Choquet integral is taken with respect to some concretepossibility measures, the pointwise estimate in terms of the modulus ofcontinuity is detailed. Some simple concrete examples of operators improvingthe classical error estimates are presented. Potential applications topractical methods dealing with data, like learning theory and regressionmodels, also are mentioned.

Quantitative estimates in approximation by Bernstein-Durrmeyer-Choquet operators with respect to monotone and submodular set functions