Approximation of classes Bpr,θ of periodic functions of one and several variables

We obtain order-sharp estimates of best approximations to the classes B-p(r),(theta) of periodic functions of several variables in the space L-q , 1 <= p, q <= infinity, by trigonometric polynomials with "numbers" of harmonics from step hyperbolic crosses. In the one-dimensional case, we establish the order of deviation of Fourier partial sums of functions from the classes B-1(r),(1)(theta) in the space L-1.