  We obtained exact order estimates of approximation of the classes
$S^{\boldsymbol{r}}_{1,\theta}B$ 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, $L_q(\mathbb{R}^d)$,
$1<q\leqslant \infty$.
