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

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  • Nazim I. Mahmudov

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).

Suggested Citation

  • Nazim I. Mahmudov, 2014. "Approximation by Genuine -Bernstein-Durrmeyer Polynomials in Compact Disks in the Case," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-11, March.
    • Handle: RePEc:hin:jnlaaa:959586
    DOI: 10.1155/2014/959586
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