  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.

Classical Analysis and ODEs