Abstract
E.V. Voronovskaya and S.N. Bernstein established an asymptotic representation for the deviation of functions from Bernstein polynomials under the condition that the function has an even-order derivative. In the present paper, a similar problem is solved in the case when the function has an odd-order derivative. In addition, analogous representations are obtained for the deviations of functions from Kantorovich polynomials.
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References
E. V. Voronovskaya, “Determination of the Asymptotic Form of the Approximation of Functions by S.N. Bernstein’s Polynomials,” Dokl. Akad. Nauk SSSR A 4, 79–85 (1932).
S. Bernstein, “Complément à l’article de E. Voronovskaya ‘Détermination de la forme asymptotique de l’approximation des fonctions par les polynômes de M. Bernstein’,” Dokl. Akad. Nauk SSSR A 4, 86–92 (1932).
G. G. Lorentz, Bernstein Polynomials (Chelsea Publ., New York, 1986).
R. A. DeVore and G. G. Lorentz, Constructive Approximation (Springer, Berlin, 1993).
K. G. Ivanov, “On Bernstein Polynomials,” C. R. Acad. Bulg. Sci. 35, 893–896 (1982).
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Original Russian Text © S.A. Telyakovskii, 2008, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Vol. 260, pp. 289–296.
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Telyakovskii, S.A. On the approximation of differentiable functions by Bernstein polynomials and Kantorovich polynomials. Proc. Steklov Inst. Math. 260, 279–286 (2008). https://doi.org/10.1134/S0081543808010197
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DOI: https://doi.org/10.1134/S0081543808010197