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Pointwise Approximation Theorems for Combinations of Bernstein Polynomials with Inner Singularities

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DOI: 10.4236/am.2011.24047    4,765 Downloads   8,924 Views  

ABSTRACT

It is well-known that Bernstein polynomials are very important in studying the characters of smoothness in theory of approximation. A new type of combinations of Bernstein operators are given in [1]. In this paper, we give the Bernstein-Markov inequalities with step-weight functions for combinations of Bernstein polynomials with inner singularities as well as direct and inverse theorems.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

W. Lu and L. Zhang, "Pointwise Approximation Theorems for Combinations of Bernstein Polynomials with Inner Singularities," Applied Mathematics, Vol. 2 No. 4, 2011, pp. 389-397. doi: 10.4236/am.2011.24047.

References

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[2] Z. Ditzian, “A Global Inverse Theorem for Combinations of Bernstein Polynomials,” Journal of Approximation Theory, Vol. 26, No. 3, 1979, pp. 277-292. doi:10.1016/0021-9045(79)90065-0
[3] Z. Ditzian and V. Totik, “Moduli of Smoothness,” Springer-Verlag, Berlin, 1987.
[4] M. Felten, “Direct and Inverse Estimates for Bernstein Polynomials,” Constructive Approximation, Vol. 14, No. 3, 1989, pp. 459-468. doi:10.1007/s003659900084
[5] S. S. Guo, C. X. Li and X. W. Liu, “Pointwise Approximation for Linear Combinations of Bernstein Operators,” Journal of Approximation Theory, Vol. 107, No. 1, 2000, pp. 109-120. doi:10.1006/jath.2000.3504
[6] G. G. Lorentz, “Bernstein Polynomial,” University of Toronto Press, Toronto, 1953.
[7] J. J. Zhang and Z. B. Xu, “Direct and Inverse Approximation Theorems with Jacobi Weight for Combinations and Higher Derivatives of Baskakov Operators,” Journal of Systems Science and Mathematical Sciences, In Chinese, Vol. 28, No. 1, 2008, pp. 30-39.
[8] D. D. Vechhia, G. Mastroianni and J. Szabados, “Weighted Approximation of Functions with Endpoint and Inner Singularities by Bernstein Operators,” Acta Mathe- matica Hungarica, Vol. 103, No. 1-2, 2004, pp. 19-41. doi:10.1023/B:AMHU.0000028234.44474.fe

  
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