In the present manuscript, we consider ?-Bernstein-Durrmeyer operators involving a strictly positive continuous function. Firstly, prove Voronovskaja type, quantitative type and Gr?ss-Voronovskaja asymptotic formula, rate of convergence by means modulus continuity for functions in Lipschitz space. Finally, show that numerical examples which describe validity theoretical example effectiveness de...

We establish a strong version of a known extremal property of Bernstein operators, as well as several characterizations of a related specific class of positive polynomial operators. © 2006 Elsevier Inc. All rights reserved. MSC: 41A36; 26D15

We investigate the iterates of positive linear operators on C[0, 1] which preserve the linear functions, and the asymptotic behaviour of the associated semigroup. The general results are applied to the Bernstein-Schnabl operators on the unit interval.

We introduce a class of Bernstein-Durrmeyer operators with respect to an arbitrary measure on a multi-dimensional simplex. These operators generalize the well-known Bernstein-Durrmeyer operators with Jacobi weights. A motivation for this generalization comes from learning theory. In the talk, we discuss the question which properties of the measure are important for convergence of the operators....

Bernstein-Nikolskii inequalities and Riesz interpolation formula are established for eigenfunctions of Laplace operators and polynomials on compact homogeneous manifolds.