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Better degree of approximation by modified Bernstein-Durrmeyer type operators

  • * Corresponding author: Ş. Y. Güngör

    * Corresponding author: Ş. Y. Güngör 
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  • In the present article we investigate a Durrmeyer variant of the generalized Bernstein-operators based on a function $ \tau(x), $ where $ \tau $ is infinitely differentiable function on $ [0, 1], \; \tau(0) = 0, \tau(1) = 1 $ and $ \tau^{\prime }(x)>0, \;\forall\;\; x\in[0, 1]. $ We study the degree of approximation by means of the modulus of continuity and the Ditzian-Totik modulus of smoothness. A Voronovskaja type asymptotic theorem and the approximation of functions with derivatives of bounded variation are also studied. By means of a numerical example, finally we illustrate the convergence of these operators to certain functions through graphs and show a careful choice of the function $ \tau(x) $ leads to a better approximation than the generalized Bernstein-Durrmeyer type operators considered by Kajla and Acar [11].

    Mathematics Subject Classification: Primary: 41A25; Secondary: 41A36.

    Citation:

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  • Figure 1.   

    Figure 2.  The convergence of $ M_{n}^{(\alpha )}(f;x) $ and $ M_{n, \tau }^{(\alpha )}(f;x) $ operators to the function $ f(x) $ for $ n = 10 $

    Figure 3.  The convergence of $ M_{n}^{(\alpha )}(f;x) $ and $ M_{n, \tau }^{(\alpha )}(f;x) $ operators to the function $ f(x) $ for $ n = 20 $

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    [2] T. AcarA. Aral and I. Raşa, Modified Bernstein-Durrmeyer operators, Gen. Math., 22 (2014), 27-41. 
    [3] A. M. AcuP. N. Agrawal and T. Neer, Approximation properties of the modified Stancu operators, Numer. Funct. Anal. Optim., 38 (2017), 279-292.  doi: 10.1080/01630563.2016.1248564.
    [4] A. M. Acu and V. A. Radu, Approximation by certain operators linking the $\alpha$- Bernstein and the genuine $\alpha$- Bernstein-Durrmeyer operators, in Mathematical Analysis I: Approximation Theory, Springer Proceedings in Mathematics & Statistics, 306 (eds. N. Deo et al.), Springer, (2020), 77-88. doi: 10.1007/978-981-15-1153-0_7.
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