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Ukrainian Mathematical Journal

, Volume 54, Issue 12, pp 1943–1957 | Cite as

On the Best Polynomial Approximations of 2π-Periodic Functions and Exact Values of n-Widths of Functional Classes in the Space L2

  • S. B. Vakarchuk
Article

Abstract

To solve extremal problems of approximation theory in the space L2, we use τ-moduli introduced by Ivanov. We determine the exact values of constants in Jackson-type inequalities and the exact values of n-widths of functional classes determined by these moduli.

Keywords

Functional Classis Approximation Theory Extremal Problem Polynomial Approximation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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REFERENCES

  1. 1.
    N. I. Chernykh, “On the best approximation of periodic functions by trigonometric polynomials in L 2,” Mat. Zametki, 2, No. 5, 513–522 (1967).Google Scholar
  2. 2.
    A. A. Ligun, “Exact Jackson-type inequalities for periodic functions in the space L 2,” Mat. Zametki, 43, No. 6, 757–769 (1988).Google Scholar
  3. 3.
    V. A. Yudin, “Diophantine approximations in extremal problems,” Dokl. Akad. Nauk SSSR, 251, No. 1, 54–57 (1980).Google Scholar
  4. 4.
    A. G. Babenko, “On an exact constant in the Jackson inequality in L 2,” Mat. Zametki, 39, No. 5, 651–664 (1986).Google Scholar
  5. 5.
    V. V. Arestov and N. I. Chernykh, “On the L 2-approximation of periodic functions by trigonometric polynomials,” in: Proceedings of the Conference “Approximation and Functions Spaces” (Gdansk, 1979), Amsterdam (1981), pp. 25–43.Google Scholar
  6. 6.
    V. Yu. Popov, “Direct and inverse inequalities for “ϕ-Fejér” mean-square approximation,” Mat. Zametki, 19, No. 3, 353–364 (1976).Google Scholar
  7. 7.
    L. V. Taikov, “Inequalities containing the best approximations and a modulus of continuity from L 2,” Mat. Zametki, 20, No. 3, 433–438 (1976).Google Scholar
  8. 8.
    L. V. Taikov, “Structural and constructive characteristics of functions from L 2,” Mat. Zametki, 25, No. 2, 217–223 (1979).Google Scholar
  9. 9.
    A. A. Ligun, “On some inequalities between the best approximations and moduli of continuity in the space L 2,” Mat. Zametki, 24, No. 6, 785–792 (1978).Google Scholar
  10. 10.
    N. Ainulloev, “The values of widths of certain classes of differentiable functions in L 2,” Dokl. Akad. Nauk Tadzh. SSR, 27, No. 8, 415–418 (1984).Google Scholar
  11. 11.
    Kh. Yussef, “Widths of classes of functions in the space L 2 (0, 2ππ),” in: Application of Functional Analysis in Approximation Theory [in Russian], Tver University, Tver (1990), pp. 167–175.Google Scholar
  12. 12.
    K. G. Ivanov, “On a new characteristic of functions. I,” Serd. Blg. Mat. Spis., 8, No. 3, 262–279 (1982).Google Scholar
  13. 13.
    K. G. Ivanov, “Direct and converse theorems for the best algebraic approximation in C [-1, 1] and L p [-1, 1],” Compt. Rend. Acad. Bulg. Sci., 33, No. 10, 1309–1312 (1980).Google Scholar
  14. 14.
    V. M. Tikhomirov, Some Problems in Approximation Theory [in Russian], Moscow University, Moscow (1976).Google Scholar
  15. 15.
    A. Pinkus, n-Widths in Approximation Theory, Springer, Berlin (1985).Google Scholar

Copyright information

© Plenum Publishing Corporation 2002

Authors and Affiliations

  • S. B. Vakarchuk
    • 1
  1. 1.Ukrainian Academy of Customs ServiceDnepropetrovsk

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