High-dimensional periodic sampling on Smolyak grids based on B-spline quasi-interpolation

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.