We describe the Besov anisotropic spaces of periodic functions of several variables in terms of the decomposition representation and establish the exact-order estimates of the Kolmogorov widths and trigonometric approximations of functions from unit balls of these spaces in the spaces L q .
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
O. V. Besov, “Investigation of one family of functional spaces in connection with the embedding and continuation theorems,” Tr. Mat. Inst. Akad. Nauk SSSR, 60, 42–81 (1961).
S. M. Nikol’skii, “Inequalities for entire functions of finite powers and their application to the theory of differentiable functions of many variables,” Tr. Mat. Inst. Akad. Nauk SSSR, 38, 244–278 (1951).
T. I. Amanov, “Representation and embedding theorems for the functional spaces S (r) p,θ B(R n ) and S (r) p,θ * B (0 ≤ x j ≤ 2π, j = 1, . . . ,n),” Tr. Mat. Inst. Akad. Nauk SSSR, 77, 5–34 (1965).
P. I. Lizorkin, “Generalized Hölder spaces B (r) p,θ and their connection with the Sobolev spaces L (r) p ,” Sib. Mat. Zh., 9, No. 5, 1127–1152 (1968).
S. M. Nikol’skii, Approximation of Functions of Many Variables and Embedding Theorems [in Russian], Nauka, Moscow (1969).
V. N. Temlyakov, Approximation of Periodic Functions, Nova Science Publishers, New York (1993).
V. N. Temlyakov, “Approximation of functions with bounded mixed derivative,” Tr. Mat. Inst. Akad. Nauk SSSR, 178, 1–112 (1986).
A. S. Romanyuk, “Approximation of the isotropic classes B r p,θ of periodic functions of many variables in the space L q ,” in: Approximation Theory of Functions and Related Problems, Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv 5, No. 1 (2008), pp. 263–278.
A. S. Romanyuk, “Approximation of the classes B r p,θ of periodic functions of one and many variables,” Mat. Zametki, 87, No. 3, 429–442 (2010).
V. V. Myronyuk, “Approximation of the classes B Ω p,θ of periodic functions of many variables by Fourier sums in the space L p with p = 1, ∞,” Ukr. Mat. Zh., 64, No. 9, 1204–1213 (2012); English translation: Ukr. Math. J., 64, No. 9, 1370–1381 (2013).
A. Kolmogoroff, “Über die beste Annäherung von Funktionen einer gegebenen Funktionenklasse,” Ann. Math., 37, No. 1, 107–110 (1936).
A. S. Romanyuk, “Bilinear approximations and Kolmogorov widths of periodic Besov classes,” in: Theory of Operators, Differential Equations, and the Theory of Functions, Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv 6, No. 1 (2009), pp. 222–236.
A. S. Romanyuk, “On the Kolmogorov and linear widths of Besov classes of periodic functions of many variables,” in: Investigations in the Approximation Theory of Functions, Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (1991), pp. 86–92.
A. S. Romanyuk, “Kolmogorov and trigonometric widths of the Besov classes B r p,θ of periodic functions of many variables,” Mat. Sb., 197, No. 1, 71–96 (2006).
A. S. Romanyuk, Approximate Characteristics of the Classes of Periodic Functions of Many Variables [in Ukrainian], Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv (2012).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 8, pp. 1117–1132, August, 2014.
Rights and permissions
About this article
Cite this article
Myronyuk, V.V. Trigonometric Approximations and Kolmogorov Widths of Anisotropic Besov Classes of Periodic Functions of Several Variables. Ukr Math J 66, 1248–1266 (2015). https://doi.org/10.1007/s11253-015-1006-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-015-1006-3