The widths of classes of analytic functions in a disc

The precise values of several $n$-widths of the classes $W^m_{p,R}(\Psi)$, $1\leqslant p<\infty$, $m\in\mathbb N$, $R\geqslant1$, in the Banach spaces $\mathscr L_{p,\gamma}$ and $B_{p,\gamma}$ are calculated, where $\gamma$ is a weight. These are classes of analytic functions $f$ in a disc of radius $R$ whose $m$th derivatives $f^{(m)}$ belong to the Hardy space $H_{p,R}$ and whose angular boundary values have averaged moduli of smoothness of second order which are majorized by the fixed function $\Psi$ on the point set $\{\pi/(2k)\}_{k\in\mathbb N}$. For the classes $W^m_{p,R}(\Psi)$ best linear methods of approximation in $\mathscr L_{p,\gamma}$ are developed. Extremal problems of related content are also considered. Bibliography: 37 titles.