Best approximations for classes of periodic functions of many variables with bounded dominating mixed derivative

We established exact in order estimates an approximation of the Sobolev classes $W_{p,\mathit{\alpha }}^{\mathit{r}}({\mathbb{T}}^d)$ of periodic functions of many variables with a bounded dominating mixed derivative. The approximation is made using trigonometric polynomials with the spectrum in step-hyperbolic crosses, and the error is estimated in the metric of the space $B_{q,1}({\mathbb{T}}^d)$, $1⩽p,q<\mathrm{\infty }$.