Approximation by Genuine qq-Bernstein-Durrmeyer Polynomials in Compact Disks in the case q>1q > 1

This paper deals with approximating properties of the newly defined $q$-generalization of the genuine Bernstein-Durrmeyer polynomials in the case $q>1$, whcih are no longer positive linear operators on $C[0,1]$. Quantitative estimates of the convergence, the Voronovskaja type theorem and saturation of convergence for complex genuine $q$-Bernstein-Durrmeyer polynomials attached to analytic functions in compact disks are given. In particular, it is proved that for functions analytic in \left\{ z\in\mathbb{C}:\left\vert z\right\vert q, the rate of approximation by the genuine $q$-Bernstein-Durrmeyer polynomials ($q>1$) is of order $q^{-n}$ versus $1/n$ for the classical genuine Bernstein-Durrmeyer polynomials. We give explicit formulas of Voronovskaja type for the genuine $q$-Bernstein-Durrmeyer for $q>1$

Approximation Theorems for Generalized Complex Kantorovich-Type Operators

The order of simultaneous approximation and Voronovskaja-type results with quantitative estimate for complex q-Kantorovich polynomials () attached to analytic functions on compact disks are obtained. In particular, it is proved that for functions analytic in , , the rate of approximation by the q-Kantorovich operators () is of order versus for the classical Kantorovich operators