Pointwise weighted approximation of functions with inner singularities by Bernstein operators
Keywords:
Pointwise weighted approximation; Bernstein operators; inner singularities.
Abstract
We consider the pointwise weighted approximation by Bernstein operators with inner singularities. The related weight functions are weights ¯ w(x) = |x − ξ|α(0 < ξ < 1, α > 0). In this paper we give direct and inverse results of this type of Bernstein polynomials.
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References
[1] H. Berens and G. Lorentz, Inverse theorems for Bernstein polynomials, Indiana Univ.Math. J. 21 (1972), pp. 693-708.
[2] D. Della Vechhia, G. Mastroianni and J. Szabados, Weighted approximation of functions with endpoint and inner singularities by Bernstein operators, Acta Math. Hungar. 103(2004), pp. 19-41.
[3] Z. Ditzian and V. Totik, Moduli of Smoothness, Springer-Verlag, Berlin, New York(1987).
[4]S.S. Guo, C.X. Li and X.W. Liu, Pointwise approximation for linear combinations of Bernstein operators, J. Approx. Theory 107 (2000), pp. 109-120.
[5] S.S. Guo, H. Tong and G. Zhang, Pointwise weighted approximation by Bernstein operators, Acta Math. Hungar. 101 (2003), pp. 293-311.
[6] G.G. Lorentz, Bernstein Polynomial, University of Toronto Press, Toronto (1953).
[7] L.S. Xie, Pointwise simultaneous approximation by combinations of Bernstein operators,J. Approx. Theory 137 (2005), pp. 1-21. rators, J. Approx. Theory 81 (1994), pp. 303-315.
[8] J.J. Zhang, Z.B. Xu, Direct and inverse approximation theorems with Jacobi weight for combinations and hihger derivatives of Baskakov operators(in Chinese), Journal of systems science and mathematical sciences. 2008 28 (1), pp. 30-39.
[9] D.X. Zhou, Rate of convergence for Bernstein operators with Jacobi weights, Acta Math.Sinica 35 (1992), pp. 331-338.
[2] D. Della Vechhia, G. Mastroianni and J. Szabados, Weighted approximation of functions with endpoint and inner singularities by Bernstein operators, Acta Math. Hungar. 103(2004), pp. 19-41.
[3] Z. Ditzian and V. Totik, Moduli of Smoothness, Springer-Verlag, Berlin, New York(1987).
[4]S.S. Guo, C.X. Li and X.W. Liu, Pointwise approximation for linear combinations of Bernstein operators, J. Approx. Theory 107 (2000), pp. 109-120.
[5] S.S. Guo, H. Tong and G. Zhang, Pointwise weighted approximation by Bernstein operators, Acta Math. Hungar. 101 (2003), pp. 293-311.
[6] G.G. Lorentz, Bernstein Polynomial, University of Toronto Press, Toronto (1953).
[7] L.S. Xie, Pointwise simultaneous approximation by combinations of Bernstein operators,J. Approx. Theory 137 (2005), pp. 1-21. rators, J. Approx. Theory 81 (1994), pp. 303-315.
[8] J.J. Zhang, Z.B. Xu, Direct and inverse approximation theorems with Jacobi weight for combinations and hihger derivatives of Baskakov operators(in Chinese), Journal of systems science and mathematical sciences. 2008 28 (1), pp. 30-39.
[9] D.X. Zhou, Rate of convergence for Bernstein operators with Jacobi weights, Acta Math.Sinica 35 (1992), pp. 331-338.
Published
2016-04-20
How to Cite
Wenming, L. (2016). Pointwise weighted approximation of functions with inner singularities by Bernstein operators. Journal of Progressive Research in Mathematics, 7(2), 982-994. Retrieved from http://scitecresearch.com/journals/index.php/jprm/article/view/484
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