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A Survey on Durrmeyer-Type Operators

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Contributions in Mathematics and Engineering

Abstract

It is well known that discretely defined operators such as Bernstein polynomials, Baskakov operators, Szász–Mirakyan, Meyer-König–Zeller operators, Stancu operators, Jain operators, and their modified versions are not possible to approximate Lebesgue integrable functions. In the year 1930, Kantorovich proposed integral modification of the Bernstein polynomials. The more general integral modification of the Bernstein polynomial was given by Durrmeyer in the year 1967. After this several Durrmeyer-type modifications of different operators have been introduced and their approximation properties have been discussed. In the present note, to the best of our knowledge, we present the different Durrmeyer-type operators introduced in the last five decades. We also propose some open problems in the end.

In Honor of Constantin Carathéodory

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Authors and Affiliations

  1. Department of Mathematics, Netaji Subhas Institute of Technology, Sector 3, Dwarka, New Delhi, 110078, India

    Vijay Gupta

  2. Department of Mathematics, National Technical University of Athens, Zografou Campus, 157 80, Athens, India

    Themistocles M. Rassias

  3. Department of Applied Mathematics, Inderprastha Engineering College, Ghaziabad, 201010, India

    Jyoti Sinha

Authors
  1. Vijay Gupta
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  2. Themistocles M. Rassias
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  3. Jyoti Sinha
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Correspondence to Vijay Gupta .

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Editors and Affiliations

  1. Industrial and Systems Engineering, University of Florida, Gainesville, Florida, USA

    Panos M. Pardalos

  2. Department of Mathematics, National Technical Unversity of Athens Department of Mathematics, Athens, Greece

    Themistocles M. Rassias

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Gupta, V., Rassias, T.M., Sinha, J. (2016). A Survey on Durrmeyer-Type Operators. In: Pardalos, P., Rassias, T. (eds) Contributions in Mathematics and Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-31317-7_14

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