2017
Том 69
№ 6

All Issues

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

Bodenchuk V. V., Serdyuk A. S.


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 $H_{h,β}$.

English version (Springer): Ukrainian Mathematical Journal 67 (2015), no. 8, pp 1137-1145.

Citation Example: Bodenchuk V. V., Serdyuk A. S. Exact Values of Kolmogorov Widths for the Classes of Analytic Functions. II // Ukr. Mat. Zh. - 2015. - 67, № 8. - pp. 1011-1018.