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Copyright © 2014 Nazim I. Mahmudov. Nazim I. Mahmudov et al. 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 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∈...:z<R , R>q , the rate of approximation by the genuine q -Bernstein-Durrmeyer polynomials q>1 is of order [superscript]q-n[/superscript] 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).

Details

Title
Approximation by Genuine q -Bernstein-Durrmeyer Polynomials in Compact Disks in the Case q>1
Author
Mahmudov, Nazim I
Publication year
2014
Publication date
2014
Publisher
John Wiley & Sons, Inc.
ISSN
10853375
e-ISSN
16870409
Source type
Scholarly Journal
Language of publication
English
ProQuest document ID
1609371108
Copyright
Copyright © 2014 Nazim I. Mahmudov. Nazim I. Mahmudov et al. 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.