Best Approximations for Classes of Periodic Functions of Many Variables with Bounded Dominating Mixed Derivative

We establish exact order estimates for the approximations of the Sobolev classes \({W}_{p,\alpha }^{r}\left({\mathbb{T}}^{d}\right)\) $W_{p,\alpha }^r\left({\mathbb{T}}^d\right)$ of periodic functions of numerous variables with bounded dominating mixed derivative. The approximations are performed by using trigonometric polynomials with spectra in step hyperbolic crosses, and the errors are estimated in the metric of the space \({B}_{q,1}\left({\mathbb{T}}^{d}\right),\) $B_{q,1}\left({\mathbb{T}}^d\right),$ 1 ≤ p, q < ∞.