Exact values of Kolmogorov widths of classes of Poisson integrals

We prove that the Poisson kernel $P_{q,\beta }(t)={\sum }_{k=1}^{\mathrm{\infty }}{q}^k\cos (kt-{\displaystyle \frac{\beta \pi }{2}})$, $q\in (0,1)$, $\beta \in \mathbb{R}$, satisfies Kushpel’s condition $C_{y,2n}$ beginning with a number $n_q$ where $n_q$ is the smallest number $n\ge 9$, for which the following inequality is satisfied: $${\displaystyle \frac{43}{10(1-q)}}{q}^{\sqrt{n}}+{\displaystyle \frac{160}{57(n-\sqrt{n})}}{\displaystyle \frac{q}{(1-q{)}^2}}\le ({\displaystyle \frac{1}{2}}+{\displaystyle \frac{2q}{(1+q^2)(1-q)}})({\displaystyle \frac{1-q}{1+q}}{)}^{\frac{4}{1-q^2}}\mathrm{.}$$ As a consequence, for all $n\ge n_q$ we obtain lower bounds for Kolmogorov widths in the space $C$ of classes $C_{\beta ,\mathrm{\infty }}^q$ of Poisson integrals of functions that belong to the unit ball in the space $L_{\mathrm{\infty }}$. 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 $C_{\beta ,\mathrm{\infty }}^q$ and show that subspaces of trigonometric polynomials of order $n-1$ are optimal for widths of dimension $2n$.