Pointwise Approximation Theorems for Combinations of Bernstein Polynomials with Inner Singularities

DOI: 10.4236/am.2011.24047   PDF   HTML     4,934 Downloads   9,106 Views   Citations


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.

Share and Cite:

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.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] D. S. Yu, “Weighted Approximation of Functions with Singularities by Combinations of Bernstein Operators,” Journal of Applied Mathematics and Computation, Vol. 206, No. 2, 2008, pp. 906-918.
[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

comments powered by Disqus

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.