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2014 Approximation by Genuine q -Bernstein-Durrmeyer Polynomials in Compact Disks in the Case q > 1
Nazim I. Mahmudov
Abstr. Appl. Anal. 2014(SI10): 1-11 (2014). DOI: 10.1155/2014/959586


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 : z < 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).


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Nazim I. Mahmudov. "Approximation by Genuine q -Bernstein-Durrmeyer Polynomials in Compact Disks in the Case q > 1 ." Abstr. Appl. Anal. 2014 (SI10) 1 - 11, 2014.


Published: 2014
First available in Project Euclid: 6 October 2014

zbMATH: 1334.41021
MathSciNet: MR3186990
Digital Object Identifier: 10.1155/2014/959586

Rights: Copyright © 2014 Hindawi

Vol.2014 • No. SI10 • 2014
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