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Relatively Uniform Weighted Summability Based on Fractional-Order Difference Operator

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Abstract

In the present paper, we introduce the notion of relatively uniform weighted summability and its statistical version based upon fractional-order difference operators of functions. The concept of relatively uniform weighted \(\alpha \beta \)-statistical convergence is also introduced and some inclusion relations concerning the newly proposed methods are derived. As an application, we prove a general Korovkin-type approximation theorem for functions of two variables and also construct an illustrative example by the help of generating function type non-tensor Meyer-König and Zeller operators. Moreover, it is shown that the proposed methods are non-trivial generalizations of relatively uniform convergence which includes a scale function. We estimate the rate of convergence of approximating positive linear operators by means of the modulus of continuity and give a Voronovskaja-type approximation theorem. Finally, we present some computational results and geometrical interpretations to illustrate some of our approximation results.

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Authors and Affiliations

  1. Department of Mathematics, Gazi University, 06500, Ankara, Turkey

    Uǧur Kadak

  2. Department of Mathematics and Statistics, University of Victoria, Victoria, BC, V8W 3R4, Canada

    H. M. Srivastava

  3. Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, 40402, Taiwan, ROC

    H. M. Srivastava

  4. Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India

    M. Mursaleen

Authors
  1. Uǧur Kadak
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  2. H. M. Srivastava
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  3. M. Mursaleen
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Correspondence to Uǧur Kadak.

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The authors declare to have no competing interests.

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Communicated by Rosihan M. Ali.

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Kadak, U., Srivastava, H.M. & Mursaleen, M. Relatively Uniform Weighted Summability Based on Fractional-Order Difference Operator. Bull. Malays. Math. Sci. Soc. 42, 2453–2480 (2019). https://doi.org/10.1007/s40840-018-0612-2

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  • DOI: https://doi.org/10.1007/s40840-018-0612-2

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