Exact values of Kolmogorov widths of classes of analytic functions

Abstract

We prove that kernels of analytic functions of kind $H_{h,β}(t)=\sum\limits_{k=1}^{\infty}\frac{1}{\cosh kh}\cos\Big(kt-\frac{βπ}{2}\Big)$, $h>0$, ${β\in\mathbb{R}}$, satisfies Kushpel's condition $C_{y,2n}$ beginning with some number $n_h$ which is explicitly expressed by parameter $h$ of smoothness of the kernel. As a consequence, for all $n\geqslant n_h$ we obtain lower bounds for Kolmogorov widths $d_{2n}$ of functional classes that are representable as convolutions of kernel $H_{h,β}$ with functions $φ\perp1$, which belong to the unit ball in the space $L_{\infty}$, in the space $C$. The obtained estimates coincide with the best uniform approximations by trigonometric polynomials for these classes. As a result, we obtain exact values for widths of mentioned classes of convolutions. Also for all $n\geqslant n_h$ we obtain exact values for Kolmogorov widths $d_{2n-1}$ of classes of convolutions of functions $φ\perp1$, which belong to the unit ball in the space $L_1$, with kernel $H_{h,β}$ in the space $L_1$.