Exact-order estimates are obtained for the best orthogonal trigonometric approximations of the Besov (B p,θ r) and Nukol’skii (H p r) classes of periodic functions of many variables in the metric of L q , 1 ≤ p, q ≤ ∞. We also establish the orders of the best approximations of functions from the same classes in the spaces L 1 and L ∞ by trigonometric polynomials with the corresponding spectrum.
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References
O. V. Besov, “Investigation of a family of functional spaces in connection with the imbedding and extension theorems,” Trudy Mat. Inst. Akad Nauk SSSR, 60, 42–61 (1961).
S. M. Nikol’skii, “Inequalities for finite-order entire functions and their application to the theory of differentiable functions of many variables,” Trudy Mat. Inst. Akad Nauk SSSR, 38, 244–278 (1951).
P. I. Lazorkin, “Generalized Hölder spaces B p,θ (r) and their relationship to the Sobolev spaces L p (r),” Sib. Mat. Zh., 9, No. 5, 1127–1152 (1968).
A. S. Romanyuk, “Approximation of the classes of periodic functions of many variables,” Mat. Zametki, 70, No. 1, 109–121 (2002).
A. S. Romanyuk, “Bilinear and trigonometric approximations of the Besov classes L p (r) of periodic functions of many variables,” Izv. Ros. Akad. Nauk, Ser. Mat., 70, No. 2, 69–98 (2006).
S. B. Stechkin, “On the absolute convergence of orthogonal series,” Dokl. Akad. Nauk SSSR, 102, No. 2, 37-40 (1955).
A. S. Romanyuk, “Best M-term trigonometric approximations of the Besov classes of periodic functions of many variables,” Izv. Ros. Akad. Nauk., Ser. Mat., 67, No. 2, 61–100 (2003).
R. A. de Vore and V. N. Temlyakov, “Nonlinear approximation by trigonometric sums,” J. Fourier Anal. Appl., 2, No. 1, 29–48 (1995).
D. Jackson, “Certain problems of closest approximation,” Bull. Amer. Math. Soc., 39, No. 12, 889–906 (1933).
N. P. Korneichuk, Exact Constants in the Approximation Theory [in Russian], Nauka, Moscow (1987).
V. N. Temlyakov, “Approximation of functions with bounded mixed derivative,” Trudy Mat. Inst. Akad. Nauk SSSR, 178, 1–112 (1986).
V. N. Temlyakov, “Greedy algorithm and m-term trigonometric approximation,” Constr. Appr., 14, 569–587 (1998).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 4, pp. 513–523, April, 2009.
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Romanyuk, A.S. Approximative characteristics of the isotropic classes of periodic functions of many variables. Ukr Math J 61, 613–626 (2009). https://doi.org/10.1007/s11253-009-0232-y
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DOI: https://doi.org/10.1007/s11253-009-0232-y