  We constructed linear algorithms of sampling recovery and cubature formulas
on Smolyak grids parametrized by $m \in \mathbb{N}$ of periodic $d$-variate
functions having Lipschitz-H\"older mixed smoothness $\alpha > 0$ based on
B-spline quasi-interpolation, and studied their optimality. We established
lower estimates (for $\alpha \le 2$) and upper bounds of the error of the
optimal sampling recovery and the optimal integration on Smolyak grids,
explicit in $d$, $m$ and the number $\nu$ of active variables of functions when
$d$ and $m$ may be large.
