Advertisement

Ukrainian Mathematical Journal

, Volume 67, Issue 8, pp 1137–1145 | Cite as

Exact Values of Kolmogorov Widths for the Classes of Analytic Functions. II

  • V. V. Bodenchuk
  • A. S. Serdyuk
Article
  • 35 Downloads

It is shown that the lower bounds of the Kolmogorov widths d 2n in the space C established in the first part of our work for the function classes that can be represented in the form of convolutions of the kernels \( {H}_{h,\beta }(t)={\displaystyle \sum_{k=1}^{\infty}\frac{1}{ \cosh kh} \cos \left(kt-\frac{\beta \pi }{2}\right),\kern1em h>0,\kern1em \beta \in \mathbb{R},} \) with functions φ ⊥ 1 from the unit ball in the space L coincide (for all nnh) with the best uniform approximations of these classes by trigonometric polynomials whose order does not exceed n − 1. As a result, we obtain the exact values of widths for the indicated classes of convolutions. Moreover, for all nnh, we determine the exact values of the Kolmogorov widths d 2n−1 in the space L 1 of classes of the convolutions of functions φ ⊥ 1 from the unit ball in the space L 1 with the kernel H h,β .

Keywords

Analytic Function Unit Ball Function Class Trigonometric Polynomial Ukrainian National Academy 
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.
    A. S. Serdyuk and V. V. Bodenchuk, “Exact values of Kolmogorov widths for the classes of analytic functions. I,” Ukr. Mat. Zh., 67, No. 6, 719–738 (2015).MathSciNetGoogle Scholar
  2. 2.
    N. I. Akhiezer, “On the best approximations of analytic functions,” Dokl. Akad. Nauk, 18, No. 4–5, 241–245 (1938).Google Scholar
  3. 3.
    S. M. Nikol’skii, “Approximations of functions by trigonometric polynomials in the mean,” Izv. Akad. Nauk SSSR, Ser. Mat., 10, 207–256 (1946).Google Scholar
  4. 4.
    A. S. Serdyuk, “On the best approximations on the classes of convolutions of periodic functions,” in: Approximation Theory of Functions and Related Problems [in Ukrainian], Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, 35 (2002), pp. 172–194.Google Scholar
  5. 5.
    A. S. Serdyuk and V. V. Bodenchuk, “Exact values of Kolmogorov widths of classes of Poisson integrals,” J. Approxim. Theory, 173, No. 9, 89–109 (2013).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Sums, Series, and Products [in Russian], Nauka, Moscow (1963).Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • V. V. Bodenchuk
    • 1
  • A. S. Serdyuk
    • 1
  1. 1.Institute of Mathematics, Ukrainian National Academy of SciencesKyivUkraine

Personalised recommendations