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

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.