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    Kopotun, K.A. Leviatan, D. Prymak, A. and Shevchuk, I.A. 2016. Yet another look at positive linear operators, <mml:math altimg="si1.gif" display="inline" overflow="scroll" xmlns:xocs="" xmlns:xs="" xmlns:xsi="" xmlns="" xmlns:ja="" xmlns:mml="" xmlns:tb="" xmlns:sb="" xmlns:ce="" xmlns:xlink="" xmlns:cals="" xmlns:sa=""><mml:mi>q</mml:mi></mml:math>-monotonicity and applications. Journal of Approximation Theory, Vol. 210, p. 1.

    Mirzaee, Farshid and Hoseini, Seyede Fatemeh 2016. Hybrid functions of Bernstein polynomials and block-pulse functions for solving optimal control of the nonlinear Volterra integral equations. Indagationes Mathematicae, Vol. 27, Issue. 3, p. 835.

    Mirzaee, Farshid Yari, Mohammad Komak and Hoseini, Seyede Fatemeh 2015. A computational method based on hybrid of Bernstein and block-pulse functions for solving linear fuzzy Fredholm integral equations system. Journal of Taibah University for Science, Vol. 9, Issue. 2, p. 252.

    Behiry, S.H. 2014. RETRACTED: Solution of nonlinear Fredholm integro-differential equations using a hybrid of block pulse functions and normalized Bernstein polynomials. Journal of Computational and Applied Mathematics, Vol. 260, p. 258.

    Behiry, S. H. 2013. Numerical Solution of Nonlinear Fredholm Integrodifferential Equations by Hybrid of Block-Pulse Functions and Normalized Bernstein Polynomials. Abstract and Applied Analysis, Vol. 2013, p. 1.

  • Bulletin of the Australian Mathematical Society, Volume 85, Issue 3
  • June 2012, pp. 353-358


  • DOI:
  • Published online: 06 February 2012

We improve the degree of pointwise approximation of continuous functions f(x) by Bernstein operators, when x is close to the endpoints of [0,1]. We apply the new estimate to establish upper and lower pointwise estimates for the test function g(x)=xlog (x)+(1−x)log (1−x). At the end we prove a general statement for pointwise approximation by Bernstein operators.

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[3]R. A. DeVore and G. G. Lorentz , Constructive Approximation (Springer, Berlin, 1993).

[4]Z. Ditzian , ‘Direct estimate for Bernstein polynomials’, J. Approx. Theory 79 (1994), 165166.

[5]Z. Ditzian and V. Totik , Moduli of Smoothness (Springer, New York, 1987).

[6]M. Felten , ‘Direct and inverse estimates for Bernstein polynomials’, Constr. Approx. 14 (1998), 459468.

[7]H. B. Knoop and X.-l. Zhou , ‘The lower estimate for linear positive operators (II)’, Res. Math. 25 (1994), 315330.

[10]V. Maier , ‘The L1-saturation class of the Kantorovich operator’, J. Approx. Theory 22 (1978), 227232.

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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