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 $[0,1{]}^d$, and ${\Phi }_n=\{{\phi }_j{\}}_{j=1}^n$ a family of $n$ functions on $[0,1{]}^d$. We consider the approximate recovery functions $f$ on $[0,1{]}^d$ from the sampled values $f(x^1),...,f(x^n)$, by the linear sampling algorithm \begin{equation} \nonumber L_n(X_n,Φ_n,f) \ := \ \sum_{j=1}^n f(x^j)φ_j. \end{equation} The error of sampling recovery is measured in the norm of the space $L_q([0,1{]}^d)$-norm or the energy norm of the isotropic Sobolev sapce $W_q^{\gamma }([0,1{]}^d)$ for $0<q\le \infty$ and $\gamma >0$. Functions $f$ to be recovered are from the unit ball in Besov type spaces of an anisotropic smoothness, in particular, spaces $B_{p,\theta }^a$ of a nonuniform mixed smoothness $a\in {\mathbb{R}}_{+}^d$, and spaces $B_{p,\theta }^{\alpha ,\beta }$ of a "hybrid" of mixed smoothness $\alpha >0$ and isotropic smoothness $\beta \in \mathbb{R}$. We constructed optimal linear sampling algorithms $L_n(X_n^{\ast },{\Phi }_n^{\ast },\cdot )$ on special sparse grids $X_n^{\ast }$ and a family ${\Phi }_n^{\ast }$ of linear combinations of integer or half integer translated dilations of tensor products of B-splines. We computed the asymptotic of the error of the optimal recovery. This construction is based on a B-spline quasi-interpolation representations of functions in $B_{p,\theta }^a$ and $B_{p,\theta }^{\alpha ,\beta }$. As consequences we obtained the asymptotic of optimal cubature formulas for numerical integration of functions from the unit ball of these Besov type spaces.

arXiv admin note: text overlap with arXiv:1009.4389

Cited by in corpus (2)

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