We obtain an exact-order estimate for the best m-term trigonometric approximation of the Besov classes \( B_{p,{{\uptheta }}}^r \) of periodic functions of many variables of low smoothness in the space L q , 1 < p ≤ 2 < q < ∞.
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References
R. A. de Vore and V. N. Temlyakov, “Nonlinear approximation by trigonometric sums,” J. Four. Anal. Appl., 2, No. 1, 29–48 (1995).
O. V. Besov, “Investigation of one family of functional spaces in connection with the theorems of imbedding and extension,” Tr. Mat. Inst. Akad. Nauk SSSR, 60, 42–61 (1961).
S. M. Nikol’skii, “Inequalities for the entire functions of finite power and their application to the theory of differentiable functions of many variables,” Tr. Mat. Inst. Akad. Nauk SSSR, 38, 244–278 (1951).
P. I. Lizorkin, “Generalized Hölder spaces \( B_{p,{{\uptheta }}}^r \) and their relations with the Sobolev spaces L p (r),” Sib. Mat. Zh., 9, No. 5, 1127–1152 (1968).
V. N. Temlyakov, “Greedy algorithm and L p (r)-term trigonometric approximation,” Constr. Approxim., 142, No. 4, 569–587 (1998).
Yanjiea Jiang and Yongping Liu, “Average widths and optimal recovery of multivariate Besov classes in L p (R d),” J. Approxim. Theory, 102, No. 1, 155–170 (2000).
A. S. Romanyuk, “Approximating characteristics of the isotropic classes of periodic functions of many variables,” Ukr. Mat. Zh., 61, No. 4, 513–523 (2009).
A. S. Romanyuk and V. S. Romanyuk, “Trigonometric and orthoprojection widths for the classes of periodic functions of many variables,” Ukr. Mat. Zh., 61, No. 10, 1348–1366 (2009).
S. B. Stechkin, “On the absolute convergence of orthogonal series,” Dokl. Akad. Nauk SSSR, 102, No. 2, 37–40 (1955).
R. S. Ismagilov, “Widths of sets in linear normed spaces and the approximation of functions by trigonometric polynomials,” Usp. Mat. Nauk, 29, No. 3, 161–178 (1974).
V. E. Maiorov, “On the linear widths of Sobolev classes,” Dokl. Akad. Nauk SSSR, 243, No. 5, 1127–1130 (1978).
B. S. Kashin, “On the approximating properties of complete orthonormal systems,” Tr. Mat. Inst. Akad. Nauk SSSR, 172, 187–191 (1985).
V. N. Temlyakov, “On the approximation of periodic functions of many variables,” Dokl. Akad. Nauk SSSR, 279, No. 2, 301–305 (1984).
É. S. Belinskii, “Approximation by a ‘floating’ system of exponents on the classes of smooth periodic functions,” Mat. Sb., 132, No. 1, 20–27 (1987).
É. S. Belinskii, “Approximation by a `floating’ system of exponents on the classes of periodic functions with bounded mixed derivative,” in: Investigations into the Theory of Functions of Many Real Variables [in Russian], Yaroslavl University, Yaroslavl (1988), pp. 16–33.
B. S. Kashin and V. N. Temlyakov, “On the best m-term approximations and the entropy of sets in the space L 1,” Mat. Zametki, 56, No. 5, 57–86 (1994).
A. S. Romanyuk, “On the best M-term trigonometric approximations for the Besov classes of periodic functions of many variables,” Izv. Ros. Akad. Nauk, Ser. Mat., 67, No. 2, 61–100 (2003).
N. P. Korneichuk, Extreme Problems in the Theory of Approximation [in Russian], Nauka, Moscow (1976).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, No. 1, pp. 104–111, January, 2010.
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Stasyuk, S.A. Best m-term trigonometric approximation for the classes \( B_{p,{{\uptheta }}}^r \) of functions of low smoothness. Ukr Math J 62, 114–122 (2010). https://doi.org/10.1007/s11253-010-0336-4
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DOI: https://doi.org/10.1007/s11253-010-0336-4