The genuine Bernstein-Durrmmeyer operator on a simplex
The modification of the Bernstein operator \(B_ n\) considered in the last few years by several authors, by incorporating the Jacobi weights \(w^{(\alpha, \beta)}\) share all the advantages of the modification made in 1967 by J. L. Durrmeyer. But these new operators lack one important property, namely the preservation of the linear functions (l.f). The author of the present paper uses the singular Jacobi weights corresponding to \(\alpha= \beta=1\). The operator considered by him combines all the nice properties of the preceding operators, including the preservation of the l.f. He investigates the extension of it to a standard simplex \(S_ m\) in \(\mathbb{R}^ m\). A special attention is given not to the approximation behaviour of the operators but to the shape preserving properties. He proves that they converge monotonically decreasing if and only if the underlying function is subharmonic with respect to the elliptic differential operator associated to the Bernstein operator or to the above-mentioned generalization of the Bernstein-Durrmeyer operators. In addition to various generalizations of convexity, subharmonicity is one further shape property being preserved. In the final part of the paper are derived, in an elementary way, pointwise and global saturation results.
scientific article; zbMATH DE number 687813
scientific article