Approximability of the classes $B_{p,\theta}^r$ of periodic functions of several variables by linear methods and best approximations

Several questions of the approximability by linear methods of the Besov classes $B_{1,\theta}^r$ and $B_{p,\theta}^r$ of periodic functions of several variables, $1\leqslant p<\infty$, are considered alongside their best approximations in the spaces $L_1$ and $L_\infty$, respectively. Taken for approximation aggregates are trigonometric polynomials with spectrum in the step hyperbolic cross. Sharp (in order) estimates of the deviations of step hyperbolic Fourier sums on the classes $B_{p,\theta}^r$, $1\leqslant p<\infty$, in the $L_\infty$ space are also obtained.