Abstract
We introduce a new type of Kantorovich–Bernstein operators. Direct and converse theorems and a Voronovskaya-type relation are given for the weighted approximation with Jacobi weights w(x)=x α(1−x)β by the new operator. None of the results involved have the restriction \({\alpha,\beta<1-\frac{1}{p}}\).
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References
B. Della Vecchia, G. Mastroianni and J. Szabados, Weighted approximation of functions with endpoint and inner singularities by Bernstein operators, Acta Math. Hungar., 103 (2004), 19–41.
B. Della Vecchia, G. Mastroianni and J. Szabados, A weighted generalization of the classical Kantorovich operator, Rend. Circ. Mat. Palermo (2), 82 (2010), 1–27.
B. Della Vecchia, G. Mastroianni and J. Szabados, A weighted generalization of the classical Kantorovich operator. II: Saturation, Mediter. J. Math., to appear.
Z. Ditzian and V. Totik, Moduli of Smoothness, Springer-Verlag (Berlin, New York, 1987).
G. G. Lorentz, Bernstein Polynomials, University of Toronto Press (Toronto, 1953).
D. X. Zhou, Rate of convergence for Bernstein operators with Jacobi weights, Acta Math. Sinica, 35 (1992), 331–338.
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Research is supported by NSF of China (10901044) and Program for excellent Young Teachers in HZNU.
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Yu, D. Weighted approximation by modified Kantorovich–Bernstein operators. Acta Math Hung 141, 132–149 (2013). https://doi.org/10.1007/s10474-013-0325-9
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DOI: https://doi.org/10.1007/s10474-013-0325-9