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On pointwise approximation properties of certain nonlinear Bernstein operators. (English) Zbl 1435.41017

Summary: The present study is concerned with the nonlinear Bernstein type operators \(NB_nf\), acting on bounded functions, where the kernel function of the operators provide some convenient assumptions. Especially, some pointwise convergence results for these type operators are achieved at a generalized Lebesgue point of the function \(f\).

MSC:

41A35 Approximation by operators (in particular, by integral operators)
41A25 Rate of convergence, degree of approximation
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References:

[1] C. Bardaro and I. Mantellini, Pointwise convergence theorems for nonlinear Mellin convolution operators, Int. J. Pure Appl. Math. 27(4) (2006), 431-447. · Zbl 1099.41013
[2] C. Bardaro, J. Musielak and G. Vinti, Nonlinear integral operators and applications, De Gruyter Series in Nonlinear Analysis and Applications, Vol. 9, xii + 201 pp., 2003. · Zbl 1030.47003
[3] G. G. Lorentz, Bernstein Polynomials, University of Toronto Press, Toronto (1953).
[4] H. Karsli, Convergence and rate of convergence by nonlinear singular integral operators depending on two parameters, Applicable Analysis, Vol. 85, Nos. 6-7, June-July 2006, 781-791. · Zbl 1110.41011
[5] H. Karsli, On approximation properties of a class of convolution type nonlinear singular integral operators, Georgian Math. Jour., Vol. 15, No. 1, (2008), 77-86. · Zbl 1144.41005
[6] H. Karsli, On approximation properties of non-convolution type nonlinear integral operators, Anal. Theory Appl., Vol. 26, No. 2, 2010, 140-152. · Zbl 1224.41070
[7] H. Karsli, On convergence of certain nonlinear Durrmeyer operators at Lebesgue points, Bulletin of the Iranian Mathematical Society, Vol. 41, No. 3, (2015), pp. 699-711. · Zbl 1373.41016
[8] H. Karsli, I. U. Tiryaki and H. E. Altin, On convergence of certain nonlinear Bernstein operators, Filomat, 30:1 (2016), 141-155. · Zbl 1474.41038
[9] H. Karsli, I. U. Tiryaki and H. E. Altin, Some approximation properties of a certain nonlinear Bernstein operators, Filomat, 28(2014), 1295-1305. · Zbl 1474.41039
[10] J. Musielak, On some approximation problems in modular spaces, In Constructive Function Theory 1981, (Proc. Int. Conf., Varna, June 1-5, 1981), pp. 455-461, Publ. House Bulgarian Acad. Sci., Sofia 1983.
[11] R. Taberski, Singular Integrals Depending on Two Parameters, Rocznicki Polskiego towarzystwa matematycznego, Seria I. Prace matematyczne, VII, 1962.
[12] S. N. Bernstein, Demonstration du Th\ueoreme de Weierstrass fond\uee sur le calcul des probabilit\ues, Comm. Soc. Math. Kharkow 13, (1912/13), 1-2.
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