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Generalized Bernstein–Durrmeyer operators of blending type

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Abstract

In this article, we present the Durrmeyer variant of generalized Bernstein operators that preserve the constant functions involving a non-negative parameter \(\rho \). We derive the approximation behaviour of these operators including a global approximation theorem via Ditzian–Totik modulus of continuity and the order of convergence for the Lipschitz type space. Furthermore, we study a Voronovskaja type asymptotic formula, local approximation theorem by means of second order modulus of smoothness and the rate of approximation for absolutely continuous functions having a derivative equivalent to a function of bounded variation. Lastly, we illustrate the convergence of these operators for certain functions using Maple software.

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Authors and Affiliations

  1. Department of Mathematics, Central University of Haryana, Mahendragarh, Haryana, 123031, India

    Arun Kajla

  2. School of Mathematics, Thapar University, Patiala, India

    Meenu Goyal

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  1. Arun Kajla
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  2. Meenu Goyal
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Correspondence to Arun Kajla.

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Kajla, A., Goyal, M. Generalized Bernstein–Durrmeyer operators of blending type. Afr. Mat. 30, 1103–1118 (2019). https://doi.org/10.1007/s13370-019-00705-z

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