Skip to main content
SpringerLink
Log in

Exact Values of the Best (𝛼, β)-Approximations for the Classes of Convolutions with Kernels that Do Not Increase the Number of Sign Changes

  • Published:
Ukrainian Mathematical Journal Aims and scope

We obtain the exact values of the best (𝛼, β) -approximations for the classes K ∗ F of periodic functions K ∗ f such that f belongs to a given rearrangement-invariant set F and K is 2𝜋 -periodic kernel that does not increase the number of sign changes by the subspaces of generalized polynomial splines with nodes at the points 2kπ/n and 2kπ/n + h,  n ∈ ℕ, k ∈ ℤ, h ∈ (0, 2π/n). It is shown that these subspaces are extremal for the Kolmogorov widths of the corresponding functional classes.

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. V. F. Babenko, “Nonsymmetric approximations in spaces of integrable functions,” Ukr. Mat. Zh., 34, No. 4, 409–416 (1982); English translation: Ukr. Math. J., 34, No. 4, 331–336 (1982).

  2. V. F. Babenko, “Inequalities for rearrangements of differentiable periodic functions, problems of approximation, and approximate integration,” Dokl. Akad. Nauk SSSR, 272, No. 5, 1038–1041 (1983).

    MathSciNet  MATH  Google Scholar 

  3. V. F. Babenko, “Approximation of the classes of convolutions,” Sib. Mat. Zh., 28, No. 5, 6–21 (1987).

    Google Scholar 

  4. V. F. Babenko, Extremal Problems of the Approximation Theory and Nonsymmetric Norms [in Russian], Doctoral-Degree Thesis (Physics and Mathematics), Dnepropetrovsk (1987).

  5. V. F. Babenko and N. V. Parfinovich, “Exact values of the best approximations of the classes of periodic functions by splines with defect 2,” Mat. Zametki, 85, No. 4, 538–551 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  6. V. F. Babenko and N. V. Parfinovich, “On the exact values of the best approximations of classes of differentiable periodic functions by splines,” Mat. Zametki, 87, No. 5, 669–683 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  7. S. Karlin, Total Positivity, Vol. I, Stanford Univ. Press, Stanford (1968).

  8. N. P. Korneichuk, Extremal Problems in the Approximation Theory [in Russian], Nauka, Moscow (1976).

    Google Scholar 

  9. N. P. Korneichuk and A. A. Ligun, “On the approximation of a class by another class and extremal subspaces in L 1 ,” Anal. Math., 7, No. 2, 107–119 (1981).

    Article  MathSciNet  Google Scholar 

  10. N. P. Korneichuk, A. A. Ligun, and V. G. Doronin, Approximation with Restrictions [in Russian], Naukova Dumka, Kiev (1982).

    Google Scholar 

  11. M. A. Krasnosel’skii and Ya. V. Rutitskii, Convex Functions and the Orlicz Spaces [in Russian], Fizmatgiz, Moscow (1958).

  12. S. G. Krein, Yu. I. Petunin, and E. M. Semenov, Interpolation of Linear Operators [in Russian], Nauka, Moscow (1978).

    Google Scholar 

  13. A. A. Ligun, “Inequalities for upper bounds of functions,” Anal. Math., 2, No. 1, 11–40 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  14. A. A. Ligun, “On the widths of some classes of differentiable periodic functions,” Mat. Zametki, 27, No. 1, 61–75 (1980).

    MathSciNet  Google Scholar 

  15. J. C. Mairhuber, I. J. Schonberg, and R. E. Williamson, “On variation diminishing transformations on the circle,” Rend. Circ. Math. Palermo, 8, No. 2, 241–270 (1959).

    Article  MathSciNet  MATH  Google Scholar 

  16. Yu. I. Makovoz, “Widths of some functional classes in the space L,” Izv. Akad. Nauk Belorus. SSR, Ser. Fiz.-Mat., 4, 19–28 (1969).

  17. Yu. I. Makovoz, “On one method for the estimation of the widths of sets in Banach spaces from below,” Mat. Sb., 87, No. 1, 136–142 (1972).

    MathSciNet  Google Scholar 

  18. Yu. I. Makovoz, “Widths of Sobolev classes and splines deviating from zero,” Mat. Zametki, 26, No. 5, 805–812 (1979).

    MathSciNet  MATH  Google Scholar 

  19. S. M. Nikol’skii, “Approximation of functions by trigonometric polynomials in the mean,” Izv. Akad. Nauk SSSR, Ser. Mat., 10, No. 3, 207–256 (1946).

  20. L. V. Taikov, “On the approximation in the mean of some classes of analytic functions,” Tr. Mat. Inst. Akad. Nauk SSSR, 88, 61–70 (1967).

    MATH  Google Scholar 

  21. A. Pinkus, “On n-widths of periodic functions,” J. Anal. Math., 35, 209–235 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  22. A. Pinkus, n-Width in Approximation Theory, Springer-Verlag, Berlin (1985).

  23. Yu. N. Subbotin, “Widths of the class W r L in L(0, 2𝜋) and the approximation by spline functions,” Mat. Zametki, 7, No. 1, 43–52 (1970).

  24. Yu. N. Subbotin, “Approximation by spline functions and estimates of the widths,” Tr. Mat. Inst. Akad. Nauk SSSR, 109, 35–60 (1971).

    MathSciNet  Google Scholar 

  25. S. P. Turovets, “On the best approximation of differentiable functions in the mean,” Dokl. Akad. Nauk Ukr. SSR, Ser. A, 5, 417–421 (1968).

Download references

Author information

Authors and Affiliations

  1. Honchar Dnipropetrovs’k National University, Dnipropetrovs’k, Ukraine

    N. V. Parfinovych

Authors
  1. N. V. Parfinovych
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 69, No. 8, pp. 1073–1083, August, 2017.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Parfinovych, N.V. Exact Values of the Best (𝛼, β)-Approximations for the Classes of Convolutions with Kernels that Do Not Increase the Number of Sign Changes. Ukr Math J 69, 1248–1261 (2018). https://doi.org/10.1007/s11253-017-1428-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-017-1428-1

Search

Navigation