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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 959586, 11 pages
http://dx.doi.org/10.1155/2014/959586
Research Article

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

Department of Mathematics, Eastern Mediterranean University, Gazimagusa, TRNC, Via Mersin 10, Turkey

Received 10 January 2014; Accepted 2 February 2014; Published 16 March 2014

Academic Editor: Sofiya Ostrovska

Copyright © 2014 Nazim I. Mahmudov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper deals with approximating properties of the newly defined -generalization of the genuine Bernstein-Durrmeyer polynomials in the case , which are no longer positive linear operators on . Quantitative estimates of the convergence, the Voronovskaja-type theorem, and saturation of convergence for complex genuine -Bernstein-Durrmeyer polynomials attached to analytic functions in compact disks are given. In particular, it is proved that, for functions analytic in , , the rate of approximation by the genuine -Bernstein-Durrmeyer polynomials is of order versus for the classical genuine Bernstein-Durrmeyer polynomials. We give explicit formulas of Voronovskaja type for the genuine -Bernstein-Durrmeyer for . This paper represents an answer to the open problem initiated by Gal in (2013, page 115).