Advertisement

Ukrainian Mathematical Journal

, Volume 66, Issue 11, pp 1731–1741 | Cite as

Best Approximations for the Cauchy Kernel on the Real Axis

  • V. M. Savchuk
  • S. O. Chaichenko
Article
  • 44 Downloads

We compute the values of the best approximations for the Cauchy kernel on the real axis ℝ by some subspaces from L q (ℝ). This result is applied to the evaluation of the sharp upper bounds for pointwise deviations of certain interpolation operators with interpolation nodes in the upper half plane and certain linear means of the Fourier series in the Takenaka–Malmquist system from the functions lying in the unit ball of the Hardy space H p , 2 ≤ p < ∞.

Keywords

Fourier Series Unit Ball Unit Disk Real Axis Hardy Space 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    V. I. Krylov, “On functions regular in a half plane,” Mat. Sb., 6, No. 1, 95–138 (1939).Google Scholar
  2. 2.
    J. Mashreghi, Representation Theorem in Hardy Spaces, Cambridge University Press, New York (2009).CrossRefGoogle Scholar
  3. 3.
    M. M. Dzhrbashyan, “Biorthogonal rational functions and the best approximation of the Cauchy kernel on the real axis,” Mat. Sb., 94, No. 3, 418–444 (1974).Google Scholar
  4. 4.
    N. I. Akhiezer, Lectures on Approximation Theory [in Russian], Nauka, Moscow (1965).Google Scholar
  5. 5.
    V. V. Savchuk, “Best linear methods of approximation and the optimal orthonormal systems of the Hardy space,” Ukr. Mat. Zh., 60, No. 5, 661–671 (2008); English translation: Ukr. Math. J., 60, No. 5, 730–743 (2008).MathSciNetCrossRefGoogle Scholar
  6. 6.
    J. B. Garnett, Bounded Analytic Functions, Academic Press, New York (1981).zbMATHGoogle Scholar
  7. 7.
    M. A. Evgrafov, Analytic Functions [in Russian], Nauka, Kiev (1991).Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • V. M. Savchuk
    • 1
  • S. O. Chaichenko
    • 2
  1. 1.Institute of MathematicsUkrainian National Academy of SciencesKyivUkraine
  2. 2.Donbass State Pedagogic UniversitySlavyanskUkraine

Personalised recommendations