We establish the exact-order estimates for the trigonometric widths of Nikol’skii–Besov \( {B}_{\infty, \theta}^r \) and Sobolev \( {W}_{\infty, \alpha}^r \) classes of periodic multivariate functions in the space L q , 1 < q < 1. The behavior of the linear widths of Nikol’skii–Besov \( {B}_{p,\theta}^r \) classes in the space L q is investigated for some relations between the parameters p and q.
Similar content being viewed by others
References
A. S. Romanyuk, “Kolmogorov and trigonometric widths of the Besov classes \( {B}_{p,\theta}^r \) of periodic functions of many variables,” Mat. Sb., 197, No. 1, 71–96 (2006).
A. S. Romanyuk, “Best approximations and widths for the classes of periodic functions of many variables,” Mat. Sb., 199, No. 2, 93–114 (2008).
A. S. Romanyuk, “Widths and the best approximation of the classes \( {B}_{p,\theta}^r \) of periodic functions of many variables,” Anal. Math., 37, 181–213 (2011).
A. S. Romanyuk, “On the problem of linear widths of the classes \( {B}_{p,\theta}^r \) of periodic functions of many variables,” Ukr. Mat. Zh., 66, No. 7, 970–982 (2014); English translation: Ukr. Math. J., 66, No. 7, 1085–1098 (2014).
A. S. Romanyuk, “Entropy numbers and widths for the classes \( {B}_{p,\theta}^r \) of periodic functions of many variables,” Ukr. Mat. Zh., 68, No. 10, 1403–1417 (2016); English translation: Ukr. Math. J., 68, No. 10, 1620–1636 (2017).
S. M. Nikol’skii Approximation of Functions of Many Variables and Embedding Theorems [in Russian], Nauka, Moscow (1969).
O. V. Besov, V. P. Il’in, and S. M. Nikol’skii, Integral Representations of Functions and Embedding Theorems [in Russian], Nauka, Moscow (1975).
T. I. Amanov, Spaces of Differentiable Functions with Predominant Mixed Derivative [in Russian], Nauka, Alma-Ata (1976).
V. N. Temlyakov, “Approximation of functions with bounded mixed derivative,” Tr. Mat. Inst. Akad. Nauk SSSR, 178, 1–112 (1986).
A. S. Romanyuk, Approximating Characteristics of the Classes of Periodic Functions of Many Variables [in Russian], Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv (2012).
D. Ding, V. N. Temlyakov, and T. Ullrich, Hyperbolic Cross Approximation, Preprint arXiv: 1601.03978 v 2[math. NA] 2 Dec. (2016).
P. I. Lizorkin and S. M. Nikol’skii, “Spaces of functions of mixed smoothness from the decomposition point of view,” Tr. Mat. Inst. Akad. Nauk SSSR, 187, 143–161 (1989).
P. S. Ismagilov, “Widths of sets in linear normalized spaces and approximation of functions by trigonometric polynomials,” Usp. Mat. Nauk, 29, No. 3, 161–178 (1974).
S. B. Stechkin, “On the absolute convergence of orthogonal series,” Dokl. Akad. Nauk SSSR, 102, No. 1, 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).
A. Kolmogoroff, “Über die beste Annäherung von Functionen einer gegeben Functionenclasse,” Ann. Math., 37, 107–111 (1936).
V. M. Tikhomirov, “Widths of sets in function spaces and the theory of best approximations,” Usp. Mat. Nauk, 15, No. 3, 81–120 (1960).
É. M. Galeev, “Linear widths of the Hölder–Nikol’skii classes of periodic functions of many variables,” Mat. Zametki, 59, No. 2, 189–199 (1996).
A. S. Romanyuk, “Approximation of the Besov classes of periodic functions of several variables in a space L q ,” Ukr. Mat. Zh., 43, No. 10, 1398–1408 (1991); English translation: Ukr. Math. J., 43, No. 10, 1297–1306 (1991).
V. N. Temlyakov, “Estimates for the asymptotic characteristics of the classes of functions with bounded mixed derivative or difference,” Tr. Mat. Inst. Akad. Nauk SSSR, 189, 138–168 (1989).
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).
A. S. Romanyuk, “Approximation of the classes \( {B}_{p,\theta}^r \) of periodic functions of many variables by linear methods and the best approximations,” Mat. Sb., 195, No. 2, 91–116 (2004).
R. A. de Vore and V. N. Temlyakov, “Nonlinear approximation by trigonometric sums,” Fourier Anal. Appl., 2, No. 1, 29–48 (1995).
Yu. V. Malykhin and K. S. Ryutin, “Product of octahedra is badly approximated in the metric of l 2,1 ,” Mat. Zametki, 101, No. 1, 85–90 (2017).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 69, No. 5, pp. 670–681, May, 2017.
Rights and permissions
About this article
Cite this article
Romanyuk, A.S. Trigonometric and Linear Widths for the Classes of Periodic Multivariate Functions. Ukr Math J 69, 782–795 (2017). https://doi.org/10.1007/s11253-017-1395-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-017-1395-6