Sampling and Cubature on Sparse Grids Based on a B-spline Quasi-Interpolation

Abstract

Let $X_n = \{x^j\}_{j=1}^n$ be a set of n points in the d-cube ${\mathbb {I}}^d:=[0,1]^d$, and $\Phi _n = \{\varphi _j\}_{j =1}^n$ a family of n functions on ${\mathbb {I}}^d$. We consider the approximate recovery of functions f on ${{\mathbb {I}}}^d$ from the sampled values $f(x^1), \ldots , f(x^n)$, by the linear sampling algorithm $L_n(X_n,\Phi _n,f) := \sum _{j=1}^n f(x^j)\varphi _j.$ The error of sampling recovery is measured in the norm of the space $L_q({\mathbb {I}}^d)$-norm or the energy quasi-norm of the isotropic Sobolev space $W^\gamma _q({\mathbb {I}}^d)$ for $1 < q < \infty$ and $\gamma > 0$. Functions f to be recovered are from the unit ball in Besov-type spaces of an anisotropic smoothness, in particular, spaces $B^{\alpha ,\beta }_{p,\theta }$ of a “hybrid” of mixed smoothness $\alpha > 0$ and isotropic smoothness $\beta \in {\mathbb {R}}$, and spaces $B^a_{p,\theta }$ of a nonuniform mixed smoothness $a \in {\mathbb {R}}^d_+$. We constructed asymptotically optimal linear sampling 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 integer translated dilations of tensor products of B-splines. We computed the asymptotic order of the error of the optimal recovery. This construction is based on B-spline quasi-interpolation representations of functions in $B^{\alpha ,\beta }_{p,\theta }$ and $B^a_{p,\theta }$. As consequences, we obtained the asymptotic order of optimal cubature formulas for numerical integration of functions from the unit ball of these Besov-type spaces.