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Let $\xi = \{x^j\}_{j=1}^n$ be a grid of $n$ points in the $d$-cube${\II}^d:=[0,1]^d$, and $\Phi = \{\phi_j\}_{j =1}^n$ a family of $n$ functionson ${\II}^d$. We define the linear sampling algorithm $L_n(\Phi,\xi,\cdot)$ foran approximate recovery of a continuous function $f$ on ${\II}^d$ from thesampled values $f(x^1), ..., f(x^n)$, by $$L_n(\Phi,\xi,f)\ := \ \sum_{j=1}^nf(x^j)\phi_j$$.  For the Besov class $B^\alpha_{p,\theta}$ of mixed smoothness $\alpha$(defined as the unit ball of the Besov space $\MB$), to study optimality of$L_n(\Phi,\xi,\cdot)$ in $L_q({\II}^d)$ we use the quantity$$r_n(B^\alpha_{p,\theta})_q \ := \ \inf_{H,\xi} \ \sup_{f \inB^\alpha_{p,\theta}} \, \|f - L_n(\Phi,xi,f)\|_q$$, where the infimum is takenover all grids $\xi = \{x^j\}_{j=1}^n$ and all families $\Phi =\{\phi_j\}_{j=1}^n$ in $L_q({\II}^d)$. We explicitly constructed linearsampling algorithms $L_n(\Phi,\xi,\cdot)$ on the grid $\xi = \ G^d(m):=\{(2^{-k_1}s_1,...,2^{-k_d}s_d) \in \II^d : \ k_1 + ... + k_d \le m\}$, with$\Phi$ a family of linear combinations of mixed B-splines which are mixedtensor products of either integer or half integer translated dilations of thecentered B-spline of order $r$. The grid $G^d(m)$ is of the size $2^m m^{d-1}$and sparse in comparing with the generating dyadic coordinate cube grid of thesize $2^{dm}$. For various $0<p,q,\theta \le \infty$ and $1/p < \alpha < r$, weproved upper bounds for the worst case error $ \sup_{f \in B^\alpha_{p,\theta}}\, \|f - L_n(\Phi,\xi,f)\|_q$ which coincide with the asymptotic order of$r_n(B^\alpha_{p,\theta})_q$ in some cases. A key role in constructing theselinear sampling algorithms, plays a quasi-interpolant representation offunctions $f \in B^\alpha_{p,\theta}$ by mixed B-spline series.

B-spline quasi-interpolant representations and sampling recovery of functions with mixed smoothness