B-Spline Quasi-Interpolation Sampling Representation and Sampling Recovery in Sobolev Spaces of Mixed Smoothness

Abstract

We proved direct and inverse theorems on B-spline quasi-interpolation sampling representation with a Littlewood-Paley-type norm equivalence in Sobolev spaces ${W^{r}_{p}}$ of mixed smoothness r. Based on this representation, we established estimates of the approximation error of recovery in L q -norm of functions from the unit ball ${U^{r}_{p}}$ in the spaces ${W^{r}_{p}}$ by linear sampling algorithms and the asymptotic optimality of these sampling algorithms in terms of Smolyak sampling width ${r^{s}_{n}}({U^{r}_{p}}, L_{q})$ and sampling width $r_{n}({U^{r}_{p}}, L_{q})$ .