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

POINTWISE APPROXIMATION BY BERNSTEIN POLYNOMIALS

  • GANCHO TACHEV (a1)
  • DOI: http://dx.doi.org/10.1017/S0004972711002838
  • Published online: 06 February 2012
Abstract
Abstract

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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