Kolmogorov widths of classes of periodic functions of one and several variables

The order of Kolmogorov widths $d_N(\widetilde W_{\bar p}^{\bar\alpha},\widetilde L_q)$ are determined for the class $\widetilde W_{\bar p}^{\bar\alpha}=\bigcap\limits_{i=1}^m\widetilde W_{p^i}^{\alpha^i}$ that is the intersection of classes of periodic functions of one variable of “higher” smoothness, in the space $\widetilde L_q$ for $1<q<\infty$, and estimates from above for “low” smoothness, and also the order of Kolmogorov widths $d_N(\widetilde H_p^r,\widetilde L_q)$ is calculated for periodic functions of several variables in the space $\widetilde L_q$ for $1<p\leqslant q\leqslant 2$. The estimate from below for $d_N(\widetilde H_p^r,\widetilde L_q)$ reduces to the estimate from below of the width of a finite-dimensional set whose width is determined.