Tbilisi Mathematical Journal

On pointwise approximation properties of certain nonlinear Bernstein operators

H. Erhan Altin

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The present study is concerned with the nonlinear Bernstein type operators $NB_{n}f$, 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 $f$.


The author is extremely grateful to the referees for their helpful remarks and valuable suggestions.

Article information

Tbilisi Math. J., Volume 12, Issue 2 (2019), 47-58.

Received: 10 August 2018
Accepted: 25 February 2019
First available in Project Euclid: 21 June 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 41A35: Approximation by operators (in particular, by integral operators)
Secondary: 41A25: Rate of convergence, degree of approximation 47G10: Integral operators [See also 45P05]

nonlinear Bernstein operators pointwise convergence generalized Lebesgue point


Altin, H. Erhan. On pointwise approximation properties of certain nonlinear Bernstein operators. Tbilisi Math. J. 12 (2019), no. 2, 47--58. doi:10.32513/tbilisi/1561082566. https://projecteuclid.org/euclid.tbilisi/1561082566

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  • C. Bardaro and I. Mantellini, Pointwise convergence theorems for nonlinear Mellin convolution operators, Int. J. Pure Appl. Math. 27(4) (2006), 431-447.
  • 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.
  • G. G. Lorentz, Bernstein Polynomials, University of Toronto Press, Toronto (1953).
  • 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.
  • 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.
  • H. Karsli, On approximation properties of non-convolution type nonlinear integral operators, Anal. Theory Appl., Vol. 26, No. 2, 2010, 140-152.
  • 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.
  • H. Karsli, I. U. Tiryaki and H. E. Altin, On convergence of certain nonlinear Bernstein operators, Filomat, 30:1 (2016), 141-155.
  • H. Karsli, I. U. Tiryaki and H. E. Altin, Some approximation properties of a certain nonlinear Bernstein operators, Filomat, 28(2014), 1295-1305.
  • 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.
  • R. Taberski, Singular Integrals Depending on Two Parameters, Rocznicki Polskiego towarzystwa matematycznego, Seria I. Prace matematyczne, VII, 1962.
  • 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.