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arXiv · 1401.2055

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

Nazim I. Mahmudov
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Abstract

This paper deals with approximating properties of the newly defined $q$-generalization of the genuine Bernstein-Durrmeyer polynomials in the case $q>1$, whcih 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 $\left\{ z\in\mathbb{C}:\left\vert z\right\vert q,$ 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 $q>1$.

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Nazim I. Mahmudov. 2014-01-09. Approximation by Genuine $q$-Bernstein-Durrmeyer Polynomials in Compact Disks in the case $q > 1$. https://arxiv.org/abs/1401.2055

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