论文标题
随机点对于Sobolev函数的近似是最佳的
Random points are optimal for the approximation of Sobolev functions
论文作者
论文摘要
我们表明，独立和均匀分布的采样点与最佳采样点一样好，可用于从有限的凸域上的sobolev space $ w_p^s（ω）$近似的近似值，$ subset \ subset \ subset \ subset \ mathbb {r}^d $ in $ l_q $ -norm If $ q <p $。更一般而言，我们表征了任意抽样点的质量$ p \ subsetω$通过$l_γ（ω）$ - 距离函数$ \ rm {dist}（\ cdot，p）$的规范，其中$γ= s（1/q-1/p）根据$ p $的覆盖半径，这对以前的特征进行了改善。
We show that independent and uniformly distributed sampling points are as good as optimal sampling points for the approximation of functions from the Sobolev space $W_p^s(Ω)$ on bounded convex domains $Ω\subset \mathbb{R}^d$ in the $L_q$-norm if $q<p$. More generally, we characterize the quality of arbitrary sampling points $P\subset Ω$ via the $L_γ(Ω)$-norm of the distance function $\rm{dist}(\cdot,P)$, where $γ=s(1/q-1/p)^{-1}$ if $q<p$ and $γ=\infty$ if $q\ge p$. This improves upon previous characterizations based on the covering radius of $P$.
