  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 $2n$.
