Exact Values of Kolmogorov Widths for the Classes of Analytic Functions. I
Abstract
We prove that the kernels of analytic functions of the form $${H}_{h,\beta }(t)={\displaystyle \sum_{k=1}^{\infty}\frac{1}{ \cosh kh} \cos \left(kt-\frac{\beta \pi }{2}\right),}h>0,\beta \in \mathbb{R},$$ satisfy Kushpel’s condition $C_{y,2n}$ starting from a certain number $n_h$ explicitly expressed via the parameter $h$ of smoothness of the kernel. As a result, for all $n ≥ n_h$ , we establish lower bounds for the Kolmogorov widths $d_{2n}$ in the space $C$ of functional classes that can be represented in the form of convolutions of the kernel $H_{h,β}$ with functions $φ⊥1$ from the unit ball in the space $L_{∞}$.Downloads
Published
25.06.2015
Issue
Section
Research articles
How to Cite
Bodenchuk, V. V., and A. S. Serdyuk. “Exact Values of Kolmogorov Widths for the Classes of Analytic Functions. I”. Ukrains’kyi Matematychnyi Zhurnal, vol. 67, no. 6, June 2015, pp. 719-38, https://umj.imath.kiev.ua/index.php/umj/article/view/2017.