Abstract
Very recently, for 0 < q < 1 Govil and Gupta [10] introduced a certain q-Durrmeyer type operators of real variable \({x \in [0,1]}\) and established some approximation properties. In the present paper, for these q-Durrmeyer operators, 0 < q < 1, but of complex variable z attached to analytic functions in compact disks, we study the exact order of simultaneous approximation and a Voronovskaja kind result with quantitative estimate. In this way, we put in evidence the overconvergence phenomenon for these q-Durrmeyer polynomials, namely the extensions of approximation properties (with quantitative estimates) from the real interval [0, 1] to compact disks in the complex plane. For q = 1 the results were recently proved in Gal-Gupta [8].
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Gal, S.G., Gupta, V. & Mahmudov, N.I. Approximation by a complex q-durrmeyer type operator. Ann Univ Ferrara 58, 65–87 (2012). https://doi.org/10.1007/s11565-012-0147-7
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DOI: https://doi.org/10.1007/s11565-012-0147-7