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[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 {z is an element of C : vertical bar z vertical bar < R}, R > 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. This paper represents an answer to the open problem initiated by Gal in (2013, page 115).