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

  • V. V. Bodenchuk
  • A. S. Serdyuk

Abstract

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 $n ≥ nh$) 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 $n ≥ nh$, 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 $H_{h,β}$.
Published
25.08.2015
How to Cite
BodenchukV. V., and SerdyukA. S. “Exact Values of Kolmogorov Widths for the Classes of Analytic Functions. II”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, no. 8, Aug. 2015, pp. 1011-8, http://umj.imath.kiev.ua/index.php/umj/article/view/2041.
Section
Research articles