Skip to main content
SpringerLink
Log in

Two-Sided Estimates for Some Functionals in Terms of the Best Approximations

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Let C be the space of continuous 2π-periodic functions. For some integrals of the form

$$ \underset{0}{\overset{\pi }{\int }}{\omega}_r\left(f,t\right)\Phi (t) dt, $$

where ω r (f, t) is the modulus of continuity of order r of a function f in C, two-sided bounds in terms of the best approximations by trigonometric polynomials are established.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. B. Stechkin, “On approximation of periodic functions by Fejér sums,” Tr. MIAN SSSR, 62, 48–60 (1961).

    MATH  Google Scholar 

  2. S. B. Stechkin, “On approximation of periodic functions by Fejér sums,” in: S. B. Stechkin, Selected Works. Mathematics [in Russian], Moscow (1989), pp. 80–91.

  3. M. F. Timan, “Some linear processes of summation of Fourier series and the best approximation,” Dokl. AN SSSR, 145, No. 4, 741–743 (1962).

    Google Scholar 

  4. M. F. Timan, “Deviation of harmonic functions from their values on the boundary and the best approximation,” Dokl. AN SSSR, 145, No. 5, 1008–1009 (1962).

    Google Scholar 

  5. M. F. Timan, “The best approximation of a function and linear methods of summation of Fourier series,” Izv. AN SSSR, Ser. Mat., 29, No. 3, 587–604 (1965).

  6. M. F. Timan, “On approximation of continuous periodic functions by linear operators based on their Fourier series,” Dokl. AN SSSR, 187, No. 6, 1339–1342 (1968).

    Google Scholar 

  7. M. F. Timan, “The best approximation of periodic functions by trigonometric polynomials and convolution type transformations,” Dokl. AN SSSR, 198, No. 4, 776–778 (1971).

    Google Scholar 

  8. M. F. Timan, Approximation and Properties of Periodic Functions [in Russian], Dnepropetrovsk (2000).

  9. V. V. Zhuk, “On approximation of a 2π-periodic function by values of a certain bounded positive operator. II,” Vestn. Leningr. Univ., Ser. Mat. Mekh. Astron., No. 13, 42–50 (1967).

  10. V. V. Zhuk, Approximation of Periodic Functions [in Russian], Leningrad (1982).

  11. G. I. Natanson, “On bounding the Lebesgue constants of Vallée-Poussin sums,” in: Geometric Questions of the Theory of Functions and Sets [in Russian], Kalinin (1986), pp. 102–108.

  12. V. V. Zhuk and G. I. Natanson, Trigonometric Fourier Series and Elements of Approximation Theory [in Russian], Leningrad (1983).

  13. A. F. Timan, Theory of Approximation of Functions of a Real Variable [in Russian], Moscow (1960).

  14. M. D. Sterlin, “Estimates of constants in inverse theorems of constructive function theory,” Dokl. AN SSSR, 209, 1296–1298 (1973).

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. St. Petersburg State University, St. Petersburg, Russia

    M. V. Babushkin & V. V. Zhuk

Authors
  1. M. V. Babushkin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. V. Zhuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. V. Babushkin.

Additional information

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 449, 2016, pp. 15–31.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Babushkin, M.V., Zhuk, V.V. Two-Sided Estimates for Some Functionals in Terms of the Best Approximations. J Math Sci 225, 848–858 (2017). https://doi.org/10.1007/s10958-017-3501-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-017-3501-6

Search

Navigation