Abstract
We obtain estimates for the ε-entropy and ε-capacity of sets of periodic functions with mean value zero that have a (ψ, β)-derivative belonging to the space L 2(0, 2π).
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REFERENCES
A. N. Kolmogorov, “On certain asymptotic characteristics of completely bounded metric spaces,” Dokl. Akad. Nauk SSSR, 108, No. 3, 385–389 (1956).
A. N. Kolmogorov and V. M. Tikhomirov, “ε-entropy and ε-capacity of sets in functional spaces,” Usp. Mat. Nauk, 14, Issue 2, 3–86 (1959).
A. I. Stepanets, Classes of Periodic Functions and Approximation of Their Elements by Fourier Sums [in Russian], Preprint No. 10, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1983).
A. I. Stepanets, “Classes of periodic functions and approximation of their elements by Fourier sums,” Dokl. Akad. Nauk SSSR, 227, No. 5, 1074–1077 (1984).
A. I. Stepanets, “Classification of periodic functions and the rate of convergence of their Fourier series,” Izv. Akad. Nauk SSSR, Ser. Mat., 50, No. 2, 101–136 (1986).
A. I. Stepanets, Classification and Approximation of Periodic Functions [in Russian], Naukova Dumka, Kiev (1987).
V. M. Tikhomirov, Some Problems in Approximation Theory [in Russian], Moscow University, Moscow (1976).
V. M. Tikhomirov, “ε-entropy and ε-capacity,” in: A. N. Kolmogorov, Information Theory and Theory of Algorithms [in Russian], Nauka, Moscow (1987), pp. 262–269.
Yu. A. Brudnyi and A. F. Timan, “Constructive characteristics of compact sets in Banach spaces and ε-entropy,” Dokl. Akad. Nauk SSSR, 126, No. 5, 927–930 (1959).
B. D. Kotlyar, “On the order of growth of ε-entropy on the class of quasismooth functions,” Usp. Mat. Nauk, 18, No. 2, 135–138 (1963).
Yu. A. Brudnyi and B. D. Kotlyar, “On the order of growth of ε-entropy on certain compact classes of functions,” Dokl. Akad. Nauk SSSR, 148, No. 5, 1001–1004 (1963).
Yu. A. Brudnyi, “On one approximation lemma and its applications,” Teor. Funkts. Funkts. Anal. Prim., Issue 16, 180–189 (1972).
A. F. Timan, Theory of Approximation of Functions of Real Variables [in Russian], Fizmatgiz, Moscow (1960).
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Kotlyar, B.D. Asymptotic Characteristics of Compact Sets and Stepanets Classes. Ukrainian Mathematical Journal 53, 229–237 (2001). https://doi.org/10.1023/A:1010460802828
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DOI: https://doi.org/10.1023/A:1010460802828