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

Abstract

This paper deals with approximating properties of the newly defined $q$-generalization of the genuine Bernstein-Durrmeyer polynomials in the case $q>1$, which are no longer positive linear operators on $C\left[\mathrm{0,1}\right]$. 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 , $R>q$, the rate of approximation by the genuine $q$-Bernstein-Durrmeyer polynomials $\left(q>1\right)$ 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 $q>1$. This paper represents an answer to the open problem initiated by Gal in (2013, page 115).