Another weighted approximation of functions with singularities by combinations of Bernstein operators

Wen-ming Lu; Jin-ping Jia

Wen-ming Lu School of Science, Hangzhou Dianzi University, Hangzhou, 310018 P.R. China
Jin-ping Jia School of Mathematics and Statistics, Tianshui Normal University, China

Keywords: Combinations of modified Bernstein polynomials, Functions with singularities, Weighted approximation, Direct and inverse results.

Abstract

We give direct and converse results for the weighted approximation of functions with inner singularities by a new type of Bernstein operators.

Downloads

Download data is not yet available.

References

P.L. Butzer, Linear combinations of Bernstein polynomials, Canad. J. Math. 5 (1953), pp. 559-567.

H. Berens and G. Lorentz, Inverse theorems for Bernstein polynomials, Indiana Univ. Math. J. 21 (1972), pp. 693-708.

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.

Z. Ditzian, A global inverse theorem for combinations of Bernstein polynomials, J. Approx. Theory 26 (1979), pp. 277-292.

Z. Ditzian and V. Totik, Moduli of Smoothness, Springer-Verlag, Berlin, New York (1987).

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.

S.S. Guo, H. Tong and G. Zhang, Pointwise weighted approximation by Bernstein operators, Acta Math. Hungar. 101 (2003), pp. 293-311.

G.G. Lorentz, Bernstein Polynomial, University of Toronto Press, Toronto (1953).

L.S. Xie, Pointwise simultaneous approximation by combinations of Bernstein operators, J. Approx. Theory 137 (2005), pp. 1-21.

L.S. Xie, The saturation class for linear combinations of Bernstein operators, Arch. Math. 91 (2008), pp. 86-96.

D.S. Yu, Weighted approximation of functions with singularities by combinations of Bernstein operators, J.Applied Mathematics and Computation. 206(2008),pp.906-918.

D.S. Yu and D.J. Zhao, Approximation of functions with singularities by truncated Bernstein operators, Southeast Bull. Math. 30 (2006), pp. 1178-1189.

D.X. Zhou, Rate of convergence for Bernstein operators with Jacobi weights, Acta Math. Sinica 35 (1992), pp. 331-338.

D.X. Zhou, On smoothness characterized by Bernstein type operators, J. Approx. Theory 81 (1994), pp. 303-315.