Approximability of the classes Bp,θr of periodic functions of several variables by linear methods and best approximations

Several questions of the approximability by linear methods of the Besov classes B1,θr and Bp,θr of periodic functions of several variables, 1≤p<∞, are considered alongside their best approximations in the spaces L1 and L∞, 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 Bp,θr, 1≤p<∞, in the L∞ space are also obtained.