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

  • V. V. Bodenchuk
  • A. S. Serdyuk

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 $L_{∞}$.
Published
25.06.2015
How to Cite
BodenchukV. V., and SerdyukA. S. “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, http://umj.imath.kiev.ua/index.php/umj/article/view/2017.
Section
Research articles