Best approximations and widths of classes of periodic functions of several variables

Order estimates are obtained for the best approximations of the Besov classes $B_{p,\theta}^r$ of periodic functions of several variables in the spaces $L_1$ and $L_\infty$ by trigonometric polynomials whose harmonic indices lie in step hyperbolic crosses. The orders of the orthoprojection widths of the classes $B_{p,\theta}^r$ and the linear widths of the classes $B_{p,\theta}^r$ and $W_{p,\alpha}^r$ in the space $L_1$ are found.
Bibliography: 22 titles.