We determine the exact values of the best (α, β)-approximations and the best one-sided approximations of classes of differentiable periodic functions by splines of defect 2. We obtain new sharp Jackson-type inequalities for the best approximations and the best one-sided approximations by splines of defect 2.
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V. F. Babenko, “Nonsymmetric approximations in spaces of summable functions,” Ukr. Mat. Zh., 34, No. 4, 409–416 (1982).
V. F. Babenko, “Nonsymmetric extremal problems in approximation theory,” Dokl. Akad. Nauk SSSR, 269, No. 3, 521–524 (1983).
N. P. Korneichuk, Exact Constants in Approximation Theory [in Russian], Nauka, Moscow (1987).
J. Favard, “Sur l’approximation des fonctions périodiques par des polynômes trigonométriques,” C. R. Acad. Sci. (Paris), 203, 1122–1124 (1936).
J. Favard, “Sur les meilleurs procédés d’approximation de certaines classes de fonctions par des polynômes trigonométriques,” Bull. Sci. Math., 61, 209–224, 243–256 (1937).
N. I. Akhiezer and M. G. Krein, “On the best approximation of differentiable periodic functions by trigonometric sums,” Dokl. Akad. Nauk SSSR, 15, 107–112 (1937).
S. M. Nikol’skii, “Approximation of functions by trigonometric polynomials in the mean,” Izv. Akad. Nauk SSSR, Ser. Mat. , 10, No. 3, 207–256 (1946).
V. M. Tikhomirov, “On n-dimensional widths of certain functional classes,” Dokl. Akad. Nauk SSSR, 130, No. 4, 734–737 (1960).
A. A. Ligun, “Inequalities for upper bounds of functions,” Ann. Math., 2, No. 1, 11–40 (1976).
T. Ganelius, “On one-sided approximation by trigonometric polynomials,” Math. Scand., 4, 247–258 (1956).
V. G. Doronin and A. A. Ligun, “Upper bounds for the best one-sided approximations by splines of the classes W r L 1,” Mat. Zametki, 19, No. 1, 11–17 (1976).
V. F. Babenko and N. V. Parfinovich, “Exact values of the best approximations of classes of periodic functions by splines of defect 2,” Mat. Zametki, 85, Issue 4, 538–551 (2009).
A. A. Ligun, V. E. Kapustyan, and Yu. I. Volkov, Special Problems of Approximation Theory and Optimal Control of Distributed Systems [in Russian], Vyshcha Shkola, Kiev (1990).
N. P. Korneichuk, A. A. Ligun, and V. G. Doronin, Approximation with Constraints [in Russian], Naukova Dumka, Kiev (1982).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 11, pp. 1443–1454, November, 2009.
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Babenko, V.F., Parfinovich, N.V. Nonsymmetric approximations of classes of periodic functions by splines of defect 2 and Jackson-type inequalities. Ukr Math J 61, 1695–1709 (2009). https://doi.org/10.1007/s11253-010-0307-9
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DOI: https://doi.org/10.1007/s11253-010-0307-9