Lower bounds for Kolmogorov widths of classes of Poisson integrals

TL;DR

This paper extends the known parameter ranges for Poisson kernels satisfying Kushpel's condition and derives exact Kolmogorov widths for classes of Poisson integrals, surpassing previous Pinkus-based methods.

Contribution

It broadens the parameter ranges for Poisson kernels meeting Kushpel's condition and provides new exact Kolmogorov width values for specific classes of Poisson integrals.

Findings

Extended the permissible $n$ ranges for Poisson kernels satisfying Kushpel's condition.

Obtained exact Kolmogorov widths for classes $C_{eta, au}^q$ in new parameter regimes.

Demonstrated results cannot be achieved using Pinkus's traditional lower bound methods.

Abstract

We expand the ranges of permissible values of $n$ ($n\in\mathbb{N}$) for which Poisson kernels $P_{q,\beta}(t)=\sum\limits_{k=1}^{\infty}q^k\cos\left(kt-\dfrac{\beta\pi}{2}\right)$, ${q\in(0,1)}$, $\beta\in\mathbb{R}$, satisfy Kushpel's condition $C_{y,2n}$. As a consequence, we obtain exact values for Kolmogorov widths in the space $C$ ($L$) of classes $C_{\beta,\infty}^q$ ($C_{\beta,1}^q$) of Poisson integrals generated by kernels $P_{q,\beta}(t)$ in new situations. It is shown that obtained here results we can't obtain by using methods of finding of exact lower bounds for widths suggested by A. Pinkus.