B-spline quasi-interpolant representations and sampling recovery of functions with mixed smoothness

Abstract

Let $\xi =\{x^j{\}}_{j=1}^n$ be a grid of $n$ points in the $d$-cube ${\II}^d:=[0,1]^d$, and $\Phi =\{{\varphi }_j{\}}_{j=1}^n$ a family of $n$ functions on ${\II}^d$. We define the linear sampling algorithm $L_n(\Phi ,\xi ,\cdot )$ for an approximate recovery of a continuous function $f$ on ${\II}^d$ from the sampled values $f(x^1),...,f(x^n)$, by $$L_n(\Phi ,\xi ,f):=\sum _{j=1}^{n}f(x^j){\varphi }_j$$. For the Besov class $B_{p,\theta }^{\alpha }$ of mixed smoothness $\alpha$ (defined as the unit ball of the Besov space $\MB$), to study optimality of $L_n(\Phi ,\xi ,\cdot )$ in $L_q({\II}^d)$ we use the quantity $$r_n(B_{p,\theta }^{\alpha }{)}_q:=\underset{H,\xi }{\inf }\underset{f\in B_{p,\theta }^{\alpha }}{\sup }\Vert f-L_n(\Phi ,xi,f){\Vert }_q$$, where the infimum is taken over all grids $\xi =\{x^j{\}}_{j=1}^n$ and all families $\Phi =\{{\varphi }_j{\}}_{j=1}^n$ in $L_q({\II}^d)$. We explicitly constructed linear sampling algorithms $L_n(\Phi ,\xi ,\cdot )$ on the grid $ξ= \ G^d(m):= \{(2^{-k_1}s_1,...,2^{-k_d}s_d) \in \II^d : \ k_1 + ... + k_d \le m\}$, with $\Phi$ a family of linear combinations of mixed B-splines which are mixed tensor products of either integer or half integer translated dilations of the centered B-spline of order $r$. The grid $G^d(m)$ is of the size $2^m{m}^{d-1}$ and sparse in comparing with the generating dyadic coordinate cube grid of the size $2^{dm}$. For various $0<p,q,\theta \le \infty$ and $1/p<\alpha <r$, we proved upper bounds for the worst case error ${\sup }_{f\in B_{p,\theta }^{\alpha }}\Vert f-L_n(\Phi ,\xi ,f){\Vert }_q$ which coincide with the asymptotic order of $r_n(B_{p,\theta }^{\alpha }{)}_q$ in some cases. A key role in constructing these linear sampling algorithms, plays a quasi-interpolant representation of functions $f\in B_{p,\theta }^{\alpha }$ by mixed B-spline series.