51. Approximation by Genuine q-Bernstein-Durrmeyer Polynomials in Compact Disks in the Case q>1.
- Author
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Mahmudov, Nazim I.
- Subjects
- *
APPROXIMATION theory , *RATIONAL root theorem , *BERNSTEIN polynomials , *POLYNOMIALS - 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 ∈ ... : |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). [ABSTRACT FROM AUTHOR]
- Published
- 2014
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