Complete asymptotic expansion for multivariate Bernstein–Durrmeyer operators and quasi-interpolants

We establish complete asymptotic expansions in terms of certain differential operators for the Bernstein–Durrmeyer operators with Jacobi weights on the dd-dimensional simplices, and for their so-called natural quasi-interpolants. Our method extensively uses spectral properties of the involved operators. In particular, we prove new identities for the eigenvalues of the operators. We also show that the obtained asymptotic expansions can be differentiated term-by-term.