Multivariate approximation by translates of the Korobov function on Smolyak grids

For a set $\mathbb{W}\subset L_p({\text{bT}}^d)$, $1<p<\mathrm{\infty }$, of multivariate periodic functions on the torus ${\text{bT}}^d$ and a given function $\varphi \in L_p(\bT^d)$,westudytheapproximationinthe$L_p(\bT^d)$-normoffunctions$f \in $\mathbb{W}$$byarbitrarylinearcombinationsof$n$translatesof$\varphi$\mathrm{.}For$$\mathbb{W}$ = U^r_p(\bT^d)$and$\varphi = \kappa_{r,d}$, we prove upper bounds of the worst case error of this approximation where $U_p^r({\text{bT}}^d)$ is the unit ball in the Korobov space $K_p^r({\text{bT}}^d)$ and ${\kappa }_{r,d}$ is the associated Korobov function. To obtain the upper bounds, we construct approximation methods based on sparse Smolyak grids. The case $p=2,r>1\mathrm{/}2$, is especially important since $K_2^r({\text{bT}}^d)$ is a reproducing kernel Hilbert space, whose reproducing kernel is a translation kernel determined by ${\kappa }_{r,d}$. We also provide lower bounds of the optimal approximation on the best choice of $\phi$.