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 $2n$.
Categories: math.CA