Approximation of Classes $B^r_{p,\theta}$ of Periodic Functions of One and Several Variables

We obtain order-sharp estimates of best approximations to the classes $B^r_{p,\theta}$ of periodic functions of several variables in the space $L_q$, $1\le p,q\le\infty$, 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 $B^{r_1}_{1,\theta}$ in the space $L_1$.