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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| Language: | English |
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Wiley
2014-01-01
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| 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). |
| format | Article |
| id | doaj-art-c99eb95d7ce540b7a4cf350005b10c40 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| 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 |