Multivariate Bernstein-Durrmeyer operators with arbitrary weight functions

Authors:
Elena E. Berdysheva;Kurt Jetter
Affiliations:
Institut für Angewandte Mathematik und Statistik, Universität Hohenheim, D-70593 Stuttgart, Germany;Institut für Angewandte Mathematik und Statistik, Universität Hohenheim, D-70593 Stuttgart, Germany
Venue:
Journal of Approximation Theory
Year:
2010

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Abstract

In this paper we introduce a class of Bernstein-Durrmeyer operators with respect to an arbitrary measure @r 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 inner product. We discuss spectral properties of the operators. We make first steps in understanding convergence of the operators.