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 C0,1. 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 z∈ℂ:zq, 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. This paper represents an answer to the open problem initiated by Gal in (2013, page 115)