We obtain order estimates for the best M-term trigonometric approximations of the classes \( B^{\Omega }_{{p,{\text{ $ \theta $ }}}} \) of periodic functions of many variables in the space L q for several values of the parameters p and q.
Similar content being viewed by others
References
S. Youngsheng and W. Heping, “Representation and approximation of multivariate periodic functions with bounded mixed moduli of smoothness,” Tr. Mat. Inst. Ros. Akad. Nauk, 219, 356–377 (1997).
N. K. Bari and S. B. Stechkin, “Best approximations and differential properties of two conjugate functions,” Tr. Mosk. Mat. Obshch., 5, 482–522 (1956).
N. N. Pustovoitov, “Representation and approximation of periodic functions of many variables with mixed modulus of continuity,” Anal. Math., 20, 35–48 (1994).
S. M. Nikol'skii, Approximation of Functions of Many Variables and Imbedding Theorems [in Russian], Nauka, Moscow (1969).
S. A. Stasyuk and O. V. Fedunyk, “Approximate characteristics of the classes {ie1416-01} of periodic functions of many variables,” Ukr. Mat. Zh., 58, No. 5, 692–704 (2006).
S. B. Stechkin, “On absolute convergence of orthogonal series,” Dokl. Akad. Nauk SSSR, 102, No. 1, 37–40 (1955).
V. N. Temlyakov, “Approximation of functions with bounded mixed derivative,” Tr. Mat. Inst. Akad. Nauk SSSR, 178, 1–112 (1986).
V. N. Temlyakov, Approximation of Periodic Functions, Nova Science, New York (1993).
É. S. Belinskii, “Approximation by a ‘floating’ system of exponents on classes of periodic functions with bounded mixed derivative,” in: Investigations in the Theory of Functions of Many Real Variables [in Russian], Yaroslavl University, Yaroslavl (1988), pp. 16–33.
A. S. Romanyuk, “Best M-term trigonometric approximations of Besov classes of periodic functions of many variables,” Izv. Ros. Akad. Nauk, Ser. Mat., 67, No. 2, 61–100 (2003).
S. M. Nikol'skii, “Inequalities for entire functions of finite order and their application to the theory of differentiable functions of many variables,” Tr. Mat. Inst. Akad. Nauk SSSR, 38, 244–278 (1951).
D. Jackson, “Certain problems of closest approximation,” Bull. Amer. Math. Soc., 39, 889–906 (1933).
O. V. Fedunyk, “Estimates for approximation characteristics of the classes {ie1416-02} of periodic functions of many variables in the space L q ,” in: Problems of Approximation Theory and Related Problems [in Ukrainian], Vol. 2, Issue 2, Institute of Mathematics, Ukrainian Academy of Sciences, Kyiv (2005), pp. 268–294.
S. A. Stasyuk, “The best M-term trigonometric approximations of the classes {ie1416-03} of functions of many variables,” Ukr. Mat. Zh., 54, No. 3, 381–394 (2002).
N. P. Korneichuk, Exact Constants in Approximation Theory [in Russian], Nauka, Moscow (1987).
A. F. Konohrai and S. A. Stasyuk, “The best orthogonal trigonometric approximations of the classes {ie1417-01} of periodic functions of many variables,” in: Problems of Approximation Theory of Functions and Related Problems [in Ukrainian], Vol. 4, Issue 1, Institute of Mathematics, Ukrainian Academy of Sciences, Kyiv (2007), pp. 151–171.
S. A. Stasyuk, “Approximation of the classes {ie1417-02} of functions of many variables in the uniform metric,” Ukr. Mat. Zh., 54, No. 11, 1551–1559 (2002).
B. S. Kashin and V. N. Temlyakov, “On the best m-term approximations and entropy of sets in the space L 1,” Mat. Zametki, 56, No. 5, 57–86 (1994).
R. A. DeVore and V. N. Temlyakov, “Nonlinear approximation by trigonometric sums,” J. Fourier Anal. Appl., 2, No. 1, 29–48 (1995).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 9, pp. 1196–1214, September, 2008.
Rights and permissions
About this article
Cite this article
Konohrai, A.F., Stasyuk, S.A. Best M-term trigonometric approximations of the classes \( B^{\Omega }_{{p,{\text{ $ \theta $ }}}} \) of periodic functions of many variables in the space L q . Ukr Math J 60, 1396–1417 (2008). https://doi.org/10.1007/s11253-009-0144-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-009-0144-x