The Genuine Bernstein{Durrmeyer Operator on a Simplex

In 1967 Durrmeyer introduced a modiication of the Bernstein polynomials as a selfadjoint polynomial operator on L 2 0; 1] which proved to be an interesting and rich object of investigation. Incorporating Jacobi weights Berens and Xu obtained a more general class of operators, sharing all the advantages of Durrmeyer's modiication, and identiied these operators as de la Vall ee{Poussin means with respect to the associated Jacobi polynomial expansion. Nevertheless, all these modiications lack one important property of the Bernstein polynomials, namely the preservation of linear functions. To overcome this drawback a Bernstein{Durrmeyer operator with respect to a singular Jacobi weight will be introduced and investigated. For this purpose an orthogonal series expansion in terms generalized Jacobi polynomials and its de la Vall ee{Poussin means will be considered. These Bernstein{Durrmeyer polynomials with respect to the singular weight combine all the nice properties of Bernstein{Durrmeyer polynomials with the preservation of linear functions, and are closely tied to classical Bernstein polynomials. Focusing not on the approximation behavior of the operators but on shape preserving properties, these operators we will prove them to converge monotonically decreasing, if and only if the underlying function is subhar-monic with respect to the elliptic diierential operator associated to the Bernstein as well as to these Bernstein{Durrmeyer polynomials. In addition to various generalizations of convexity, subharmonicity is one further shape property being preserved by these Bernstein{Durrmeyer polynomials. Finally, pointwise and global saturation results will be derived in a very elementary way.