Constructive Approximation

, Volume 13, Issue 1, pp 17–27 | Cite as

Exactn-widths of hardy-sobolev classes

  • K. Y. Osipenko


Let\(\tilde h^r _{\infty ,\beta } \) and\(\tilde H^r _{\infty ,\beta } \) denote those 2π-periodic, real-valued functions onR that are analytic in the strip |Imz|<β and satisfy the restrictions |Ref (r)(z)| ≤ 1 and |f (r)(z)| ≤ 1, respectively. We determine the Kolmogorov, linear, and Gel’fand widths of\(\tilde h^r _{\infty ,\beta } \) inL q[0, 2π], 1 ≤q ≤ ∞, and\(\tilde H^r _{\infty ,\beta } \) inL [0, 2π].

AMS classification

41A46 30D55 

Key words and phrases

Hardy-Sobolev class n-Width Bounded analytic functions 


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  1. 1.
    N. I. Akhiezer (1965): Lectures on the Theory of Approximation. Moscow: Nauka.Google Scholar
  2. 2.
    N. I. Akhiezer (1970): Elements of the Theory of Elliptic Functions. Moscow: Nauka.Google Scholar
  3. 3.
    Yu. A. Farkov (1993):The N-widths of Hardy-Sobolev spaces of several complex variables. J. Approx. Theory,75:183–197.zbMATHCrossRefGoogle Scholar
  4. 4.
    S. D. Fisher (1989):Envelopes, widths, and Landau problems for analytic functions. Constr. Approx.5:171–187.zbMATHCrossRefGoogle Scholar
  5. 5.
    S. Ya. Khavinson (1985): Two Papers on Extremal Problems in Complex Analysis. Amer. Math. Soc. Translations, Ser. 2, No. 129.Google Scholar
  6. 6.
    N. P. Korneichuk (1987): Exact Constants in the Theory of Approximation. Moscow: Nauka.Google Scholar
  7. 7.
    K. Yu. Osipenko (1994):On n-widths, optimal quadrature formulas, and optimal recovery of functions analytic in strip. Izv. Ross. Akad. Nauk Ser. Mat.58:55–79; English translation in Russian Acad. Sci. Izv. Math.45:55–78.zbMATHGoogle Scholar
  8. 8.
    K. Yu. Osipenko (1995):On N-widths of holomorphic functions of several variables. J. Approx. Theory,82:135–155.zbMATHCrossRefGoogle Scholar
  9. 9.
    K. Yu. Osipenko (1995):Exact values of n-widths and optimal quadratures on classes of bounded analytic and harmonic functions. J. Approx. Theory,82:156–175.zbMATHCrossRefGoogle Scholar
  10. 10.
    A. Pinkus (1985):n-Widths in Approximation Theory. Berlin: Springer-Verlag.zbMATHGoogle Scholar
  11. 11.
    V. M. Tikhomirov (1960):Diameters of sets in function spaces and the theory of best approximations. Uspekhi Mat. Nauk,15:81–120; English translation in Russian Math. Surveys,15:75–111.Google Scholar

Copyright information

© Springer-Verlag New York Inc 1997

Authors and Affiliations

  • K. Y. Osipenko
    • 1
  1. 1.Department of MathematicsMoscow State University of Aviation TechnologyMoscowRussia

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