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Mathematical Notes

, Volume 93, Issue 5–6, pp 917–922 | Cite as

Sharp constant in Jackson’s inequality with modulus of smoothness for uniform approximations of periodic functions

  • S. A. PichugovEmail author
Article
  • 53 Downloads

Abstract

It is proved that, in the space C, for all k, n ∈ ℕ,n > 1, the following inequalities hold:
where e n−1(f) is the value of the best approximation of f by trigonometric polynomials and ω 2(f, h) is the modulus of smoothness of f. A similar result is also obtained for approximation by continuous polygonal lines with equidistant nodes.

Keywords

Jackson’s inequality periodic function trigonometric polynomial modulus of smoothness polygonal line Steklov mean Favard sum 

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References

  1. 1.
    N. P. Korneichuk, “The exact constant in Jackson’s theorem on best uniform approximation of continuous periodic functions,” Dokl.Akad. Nauk SSSR 145(3), 514–515 (1962) [Soviet Math. Dokl. 3 (3), 1040–1041 (1962)].MathSciNetGoogle Scholar
  2. 2.
    N. P. Korneichuk, “On the sharp constant in Jackson’s inequality for continuous periodic functions,” Mat. Zametki 32(5), 669–674 (1982) [Math. Notes 32 (5), 818–821 (1983)].MathSciNetGoogle Scholar
  3. 3.
    N. I. Chernykh, “On Jackson’s inequality in L 2,” in Trudy Mat. Inst. Steklov, Vol. 88: Approximation of Functions in the Mean, Collection of papers (Nauka, Moscow, 1967), pp. 71–74 [Proc. Steklov Inst. Math. 88, 75–78 (1967)].Google Scholar
  4. 4.
    N. I. Chernykh, “Best approximation of periodic functions by trigonometric polynomials in L 2,” Mat. Zametki 2(5), 513–522 (1967) [Math. Notes 2 (5), 803–808 (1968)].MathSciNetGoogle Scholar
  5. 5.
    N. I. Chernykh, “Jackson’s inequality in L p(0, 2π), (1 ≤ p < 2), with sharp constant,” in Trudy Mat. Inst. Steklov, Vol. 198: Proceedings of an All-Union School on the Theory of Functions, Miass, July 1989 (Nauka, Moscow, 1992), pp. 232–241 [Proc. Steklov Inst. Math. 198, 223–231 (1994)].Google Scholar
  6. 6.
    V. V. Shalaev “On the approximation of continuous periodic functions by trigonometric polynomials,” in Studies of Problems of Current Interest Dealing with Summation and Approximations of Functions and Their Applications (Dnepropetrovsk. Univ., Dnepropetrovsk, 1979), pp. 39–43 [in Russian].Google Scholar
  7. 7.
    N. P. Korneichuk, Exact Constants in Approximation Theory (Nauka, Moscow, 1987) [in Russian].Google Scholar
  8. 8.
    A. A. Ligun, “Sharp constants in inequalities of Jackson type,” in Special Questions of Approximation Theory and Optimal Control of Distributed Systems (Vishcha Shkola, Kiev, 1990), pp. 3–74 [in Russian].Google Scholar
  9. 9.
    V. V. Zhuk, “Some sharp inequalities between best approximations and moduli of continuity,” Vestnik Leningrad. Univ. Mat. Mekh. Astronom., No. 1, 21–26 (1974).Google Scholar
  10. 10.
    V. V. Zhuk, Approximation of Periodic Functions (Izd. Leningradsk. Univ., Leningrad, 1982) [in Russian].zbMATHGoogle Scholar
  11. 11.
    A. G. Babenko and Yu. V. Kryakin, “Integral approximation of the characteristic function of the interval and Jackson’s inequality in C(\( \mathbb{T} \)),” Trudy Inst. Mat. Mekh. (Ural Branch, Russian Academy of Sciences) (2009), Vol. 15, pp. 59–65 [in Russian].Google Scholar
  12. 12.
    N. P. Korneichuk, Splines in Approximation Theory (Nauka, Moscow, 1984) [in Russian].Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  1. 1.Dnepropetrovsk National Technical University of Railroad CommunicationsKievRussia

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