Lower bounds for Kolmogorov widths of classes of convolutions with Neumann kernel

TL;DR

This paper derives exact lower bounds for Kolmogorov widths of convolution classes with Neumann kernels, matching best uniform approximations, thus providing precise width values for these function classes.

Contribution

The paper establishes exact lower bounds for Kolmogorov widths of convolution classes with Neumann kernels, matching best uniform approximations, and determines their exact values.

Findings

Exact lower bounds for Kolmogorov widths are obtained.

Bounds coincide with best uniform trigonometric polynomial approximations.

Exact widths of the classes are explicitly determined.

Abstract

We obtain exact lower bounds for Kolmogorov $n$-widths in spaces $C$ and $L$ of classes of convolutions with Neumann kernel $N_{q,\beta}(t)=\sum\limits_{k=1}^{\infty}\dfrac{q^k}{k}\cos\left(kt-\dfrac{\beta\pi}{2}\right)$, ${q\in(0,1)}$, ${\beta\in\mathbb{R}}$, for all natural $n$ greater some number which depend only on $q$. The obtained estimates coincide with the best uniform approximations by trigonometric polynomials of mentioned classes. It made possible to obtain exact values for widths of these classes.