Abstract
We obtain estimates exact in order for the best approximations of the classes B r∞,θ of periodic functions of two variables in the metric of L ∞ by trigonometric polynomials whose spectrum belongs to a hyperbolic cross. We also investigate the best approximations of the classes B r p,θ , 1 ≤ p < ∞, of periodic functions of many variables in the metric of L ∞ by trigonometric polynomials whose spectrum belongs to a graded hyperbolic cross.
Similar content being viewed by others
References
A. S. Romanyuk, “Approximation of the classes B r p,θ of periodic functions of many variables by linear methods and best approximations,” Mat. Sb., 195, No. 2, 91–116 (2004).
O. V. Besov, “On one family of functional spaces. Theorems on imbedding and extension,” Dokl. Akad. Nauk SSSR, 126, No. 6, 1163–1165 (1959).
S. M. Nikol’skii, Approximation of Functions of Many Variables and Imbedding Theorems [in Russian], Nauka, Moscow (1969).
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).
V. N. Temlyakov, “Approximation of functions with bounded mixed derivative,” Tr. Mat. Inst. Akad. Nauk SSSR, 178, 1–112 (1986).
N. P. Korneichuk, Exact Constants in Approximation Theory [in Russian], Nauka, Moscow (1987).
A. S. Romanyuk, “Kolmogorov widths of the Besov classes B r p,θ in the metric of the space L ∞,” Ukr. Mat. Visn., 2, No. 2, 201–218 (2005).
Author information
Authors and Affiliations
Additional information
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 10, pp. 1395–1406, October, 2006.
Rights and permissions
About this article
Cite this article
Romanyuk, A.S. Best approximations of the classes B r p,θ of periodic functions of many variables in uniform metric. Ukr Math J 58, 1582–1596 (2006). https://doi.org/10.1007/s11253-006-0155-9
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s11253-006-0155-9