Exact values of Kolmogorov widths of classes of analytic functions

TL;DR

This paper derives exact values for Kolmogorov widths of classes of analytic functions defined via specific kernels, using Kushpel's condition, and matches these with best uniform approximations.

Contribution

It establishes explicit conditions under which the Kolmogorov widths of certain convolution classes are exactly determined, extending previous approximation theory results.

Findings

Exact values of Kolmogorov widths for classes of analytic functions are obtained.

Kushpel's condition is verified for specific kernels, enabling precise width calculations.

The results match the best uniform approximations by trigonometric polynomials.

Abstract

We prove that kernels of analytic functions of kind $H_{h,\beta}(t)=\sum\limits_{k=1}^{\infty}\frac{1}{\cosh kh}\cos\Big(kt-\frac{\beta\pi}{2}\Big)$, $h>0$, ${\beta\in\mathbb{R}}$, 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 $H_{h,\beta}$ with functions $\varphi\perp1$, 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 $\varphi\perp1$, which belong to the unit ball in the space $L_1$, with kernel $H_{h,\beta}$ in the space $L_1$.