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Let $X_n = \{x^j\}_{j=1}^n$ be a set of $n$ points in the $d$-cube $[0,1]^d$,and $\Phi_n = \{\varphi_j\}_{j =1}^n$ a family of $n$ functions on $[0,1]^d$.We consider the approximate recovery functions $f$ on $[0,1]^d$ from thesampled values $f(x^1), ..., f(x^n)$, by the linear sampling algorithm\begin{equation} \nonumber L_n(X_n,\Phi_n,f) \ := \ \sum_{j=1}^nf(x^j)\varphi_j. \end{equation} The error of sampling recovery is measured inthe norm of the space $L_q([0,1]^d)$-norm or the energy norm of the isotropicSobolev sapce $W^\gamma_q([0,1]^d)$ for $0 < q \le \infty$ and $\gamma > 0$.Functions $f$ to be recovered are from the unit ball in Besov type spaces of ananisotropic smoothness, in particular, spaces $B^a_{p,\theta}$ of a nonuniformmixed smoothness $a \in {\mathbb R}^d_+$, and spaces$B^{\alpha,\beta}_{p,\theta}$ of a "hybrid" of mixed smoothness $\alpha > 0$and isotropic smoothness $\beta \in \mathbb R$. We constructed optimal linearsampling algorithms $L_n(X_n^*,\Phi_n^*,\cdot)$ on special sparse grids $X_n^*$and a family $\Phi_n^*$ of linear combinations of integer or half integertranslated dilations of tensor products of B-splines. We computed theasymptotic of the error of the optimal recovery. This construction is based ona B-spline quasi-interpolation representations of functions in $B^a_{p,\theta}$and $B^{\alpha,\beta}_{p,\theta}$. As consequences we obtained the asymptoticof optimal cubature formulas for numerical integration of functions from theunit ball of these Besov type spaces.

Sampling and cubature on sparse grids based on a B-spline quasi-interpolation