Exact values of Kolmogorov widths of classes of Poisson integrals

TL;DR

This paper derives exact Kolmogorov widths for classes of Poisson integrals, establishing optimal approximation subspaces and confirming the best uniform approximation by trigonometric polynomials.

Contribution

It provides explicit formulas for Kolmogorov widths of Poisson integral classes and identifies optimal polynomial subspaces for approximation.

Findings

Exact Kolmogorov widths for classes of Poisson integrals are obtained.

Subspaces of trigonometric polynomials are proven to be optimal for these widths.

The results match the best uniform approximations by trigonometric polynomials.

Abstract

We prove that the Poisson kernel $P_{q,\beta}(t)=\sum\limits_{k=1}^{\infty}q^k\cos(kt-\dfrac{\beta\pi}{2})$, ${q\in(0,1)}$, $\beta\in\mathbb{R}$, satisfies Kushpel's condition $C_{y,2n}$ beginning with a number $n_q$ where $n_q$ is the smallest number $n\geq9$, for which the following inequality is satisfied: $$ \dfrac{43}{10(1-q)}q^{\sqrt{n}}+\dfrac{160}{57(n-\sqrt{n})}\; \dfrac{q}{(1-q)^2}\leq (\dfrac{1}{2}+\dfrac{2q}{(1+q^2)(1-q)})(\dfrac{1-q}{1+q})^{\frac {4}{1-q^2}}. $$ As a consequence, for all $n\geq n_q$ we obtain lower bounds for Kolmogorov widths in the space $C$ of classes $C_{\beta,\infty}^q$ of Poisson integrals of functions that belong to the unit ball in the space $L_\infty$. 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 classes $C_{\beta,\infty}^q$ and show that subspaces of trigonometric polynomials of order $n-1$ are optimal for widths of dimension $2n$.