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Approximation by \(q\)-Durrmeyer type polynomials in compact disks in the case \(q > 1\). (English) Zbl 1334.41021

Summary: Recently, R. P. Agarwal and V. Gupta [J. Inequal. Appl. 2012, Article ID 111, 13 p. (2012; Zbl 1273.30025)] studied some approximation properties of the complex \(q\)-Durrmeyer type operators in the case \(0 < q < 1\). In this paper this study is extended to the case \(q > 1\). More precisely, approximation properties of the newly defined generalization of this operators in the case \(q > 1\) are studied. Quantitative estimates of the convergence, the Voronovskaja type theorem and saturation of convergence for complex \(q\)-Durrmeyer type polynomials attached to analytic functions in compact disks are given. In particular, it is proved that for functions analytic in \(\left\{z \in \mathbb{C} : \mid z \mid < R\right\}, R > q\), the rate of approximation by the \(q\)-Durrmeyer type polynomials \((q > 1)\) is of order \(q^{- n}\) versus \(1 / n\) for the classical \((q = 1)\) Durrmeyer type polynomials. Explicit formulas of Voronovskaya type for the \(q\)-Durrmeyer type operators for \(q > 1\) are also given. This paper represents an answer to the open problem initiated by S. G. Gal [Overconvergence in complex approximation. New York, NY: Springer (2013; Zbl 1272.30001)].

MSC:

41A35 Approximation by operators (in particular, by integral operators)
05A30 \(q\)-calculus and related topics
30E10 Approximation in the complex plane
33D05 \(q\)-gamma functions, \(q\)-beta functions and integrals
41A10 Approximation by polynomials
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