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

Let Xn = {x j }j=1 be a set of n points in the d-cube Id := [0, 1]d , and n = {φ j }j=1 a family of n functions on Id . We consider the approximate recovery of functions f on Id from the sampled values f (x1), . . . , f (xn), by the linear sampling algorithm Ln(Xn, n, f ) := ∑nj=1 f (x j )φ j . The error of sampling recovery is measured in the norm of the space Lq(I)-norm or the energy quasi-norm of the isotropic Sobolev space W γ q (Id) for 1 < q < ∞ and γ > 0. Functions f to be recovered are from the unit ball in Besov-type spaces of an anisotropic smoothness, in particular, spaces B p,θ of a “hybrid” of mixed smoothness α > 0 and isotropic smoothness β ∈ R, and spaces Bp,θ of a nonuniform mixed smoothness a ∈ R+. We constructed asymptotically optimal linear sampling algorithms Ln(X∗ n, ∗n, ·) on special sparse grids X∗ n and a family ∗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 p,θ and B a p,θ . 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.