Abstract
As an important type of polynomial approximation, approximation of functions by Bernstein operators is an important topic in approximation theory and computational theory. This paper gives global and pointwise estimates for weighted approximation of functions with singularities by Bernstein operators. The main results are the Jackson’s estimates of functions f ∈(W w,λ)2 and f ∈ C w , which extends the result of (Della Vecchia et al., 2004).
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Della Vecchia, B., Mastroianni, G., Szabodos, J., 2004. Weighted approximation of functions with endpoint or inner singularities by Bernstein operators. Acta Math.Hungar., 103:19–41. [doi:10.1023/B:AMHU.0000028234.44474.fe]
Ditzian, Z., Totik, V., 1987. Moduli of Smoothness. Springer-Verlag, New York, p.56.
Guo, S., Tong, H., Zhang, G., 2003. Pointwise weighted approximation by Bernstein operators. Acta Math. Hungar., 101(4):293–311. [doi:10.1023/B:AMHU.0000004941.98294.8b]
Lorentz, G.G., 1953. Bernstein Polynomials. University of Toronto Press, Toronto, p.15.
Mastroianni, G., Totik, V., 1998. Jackson type inequalities for doubling and A p weights. Rend. Circ. Mat. Palermo (II) Suppl., 52:83–99.
Mastroianni, G., Totik, V., 1999. Jackson type inequalities for doubling and A p weights II. East J. Approx., 5:101–116.
Mastroianni, G., Totik, V., 2001. Best approximation and moduli of smoothness for doubling weights. J. Approx. Theory, 110(2):180–199. [doi:10.1006/jath.2000.3546]
Yu, D.S., Zhao, D.J., 2006. Approximation of function with singularities by truncated Bernstein operators. Southeast Asian Bull. Math., 30:1169–1178.
Yu, D.S., Zhou, S.P. Global and pointwise estimate for approximation by rational functions with polynomials of positive coefficients as the denominators. in press.
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Wei, Br., Zhao, Y. Weighted approximation of functions with singularities by Bernstein operators. J. Zhejiang Univ. Sci. A 9, 1451–1456 (2008). https://doi.org/10.1631/jzus.A0820015
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DOI: https://doi.org/10.1631/jzus.A0820015