Approximation by Genuine q-Bernstein-Durrmeyer Polynomials in Compact Disks in the Case q>1

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 c...

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Main Author: Nazim I. Mahmudov
Format: Article
Language:English
Published: Wiley 2014-01-01
Series:Abstract and Applied Analysis
Online Access:http://dx.doi.org/10.1155/2014/959586
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author Nazim I. Mahmudov
author_facet Nazim I. Mahmudov
author_sort Nazim I. Mahmudov
collection DOAJ
description 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: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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spelling doaj-art-c99eb95d7ce540b7a4cf350005b10c402025-02-03T07:25:44ZengWileyAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/959586959586Approximation by Genuine q-Bernstein-Durrmeyer Polynomials in Compact Disks in the Case q>1Nazim I. Mahmudov0Department of Mathematics, Eastern Mediterranean University, Gazimagusa, TRNC, Via Mersin 10, TurkeyThis 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: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).http://dx.doi.org/10.1155/2014/959586
spellingShingle Nazim I. Mahmudov
Approximation by Genuine q-Bernstein-Durrmeyer Polynomials in Compact Disks in the Case q>1
Abstract and Applied Analysis
title Approximation by Genuine q-Bernstein-Durrmeyer Polynomials in Compact Disks in the Case q>1
title_full Approximation by Genuine q-Bernstein-Durrmeyer Polynomials in Compact Disks in the Case q>1
title_fullStr Approximation by Genuine q-Bernstein-Durrmeyer Polynomials in Compact Disks in the Case q>1
title_full_unstemmed Approximation by Genuine q-Bernstein-Durrmeyer Polynomials in Compact Disks in the Case q>1
title_short Approximation by Genuine q-Bernstein-Durrmeyer Polynomials in Compact Disks in the Case q>1
title_sort approximation by genuine q bernstein durrmeyer polynomials in compact disks in the case q 1
url http://dx.doi.org/10.1155/2014/959586
work_keys_str_mv AT nazimimahmudov approximationbygenuineqbernsteindurrmeyerpolynomialsincompactdisksinthecaseq1