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

Nazim I. Mahmudov

Abstract and Applied Analysis, 2014, vol. 2014, issue 1

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 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 ∈ ℂ : |z| 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).

Date: 2014
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https://doi.org/10.1155/2014/959586

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnlaaa:v:2014:y:2014:i:1:n:959586

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