Order estimates of approximative characteristics of functions from classes $S_{1,\theta }^{r}B({\mathbb{R}}^d)$

We obtained exact order estimates of approximation of the classes $S_{1,\theta }^{\mathit{r}}B$ by entire functions of exponential type with supports of their Fourier transforms in step hyperbolic cross. The error of the approximation estimated in the metric of Lebesgue spaces, $L_q({\mathbb{R}}^d)$, $1<q⩽\mathrm{\infty }$.