Best approximation methods and widths for some classes of functions in $H_{q,\rho}$, $1\le q\le\infty$, $0<\rho\le1$

We compute the exact values of widths for various widths for the classes $W_{q,a}^{(r)}(\Phi,\mu)$, $\mu\ge1$, of analytic functions in the disk belonging to the Hardy space $H_q$, $q\ge1$, whose averaged moduli of continuity of the boundary values of the derivatives with respect to the argument $f_a^{(r)}$, $r\in\mathbb N$, are dominated by a given function $\Phi$. For calculating the linear and Gelfand $n$-widths, we use best linear approximation for these functions.