Elena E. Berdysheva (University of Hohenheim):
Multivariate Bernstein-Durrmeyer Operators with Arbitrary Weight Functions
Abstract: We introduce a class of Bernstein-Durrmeyer
operators with respect to an arbitrary measure $\rho$ on the
$d$-dimensional simplex, and a class of more general polynomial
integral operators with a kernel function involving the Bernstein
basis polynomials. These operators generalize the well-known
Bernstein-Durrmeyer operators with respect to Jacobi weights. We
investigate properties of the new operators. In particular, we study
the associated reproducing kernel Hilbert space and show that the
Bernstein basis functions are orthogonal in the corresponding scalar
product. We discuss spectral properties of the operators. We make
first steps in understanding convergence of the operators. Joint work
with Kurt Jetter.