标题： 函数混合光滑性的加权采样恢复 显示英文标题

标题： Weighted sampling recovery of functions with mixed smoothness

摘要： 我们研究稀疏网格线性采样算法及其在从$\mathbb{R}^d$的一组$n$的采样值中近似恢复具有混合光滑性的函数时的最优性，在两种不同情况下：(i) 要恢复的函数属于混合光滑性的加权 Sobolev 空间$W^r_{p,w}(\mathbb{R}^d)$，近似误差由加权 Lebesgue 空间$L_{q,w}(\mathbb{R}^d)$的范数来衡量，以及 (ii) 要恢复的函数属于具有测度$W^r_p(\mathbb{R}^d; \mu_w)$的混合光滑性的 Sobolev 空间，近似误差由具有测度$L_q(\mathbb{R}^d; \mu_w)$的 Lebesgue 空间范数来衡量。 这里，函数$w$是张量积形式的弗洛伊德型权函数，是在情形 (i) 中的权函数，而$\mu_w$是情形 (ii) 中测度的密度函数。在线性采样算法的最优性方面，根据相关的采样$n$-宽度进行了研究。我们构造了稀疏网格线性采样算法，这些算法在情形 (i) 和 (ii) 中完全不同，并给出了相应采样$n$-宽度的上界。我们证明了在一维情况下，这些算法实现了采样宽度的正确收敛速度。在高维情形 (ii)（$d\ge 2$）中，我们也通过一种非构造性方法实现了$n$-宽度的正确收敛速度，对于$1\le q \le 2 \le p \le \infty$。 显示英文摘要

摘要： We study sparse-grid linear sampling algorithms and their optimality for approximate recovery of functions with mixed smoothness on $\mathbb{R}^d$ from a set of $n$ their sampled values in two different settings: (i) functions to be recovered are in weighted Sobolev spaces $W^r_{p,w}(\mathbb{R}^d)$ of mixed smoothness and the approximation error is measured by the norm of the weighted Lebesgue space $L_{q,w}(\mathbb{R}^d)$, and (ii) functions to be recovered are in Sobolev spaces with measure $W^r_p(\mathbb{R}^d; \mu_w)$ of mixed smoothness and the approximation error is measured by the norm of the Lebesgue space with measure $L_q(\mathbb{R}^d; \mu_w)$. Here, the function $w$, a tensor-product Freud-type weight is the weight in the setting (i), and the density function of the measure $\mu_w$ in the setting (ii). The optimality of linear sampling algorithms is investigated in terms of the relevant sampling $n$-widths. We construct sparse-grid linear sampling algorithms which are completely different for the settings (i) and (ii) and which give upper bounds of the corresponding sampling $n$-widths. We prove that in the one-dimensional case, these algorithms realize the right convergence rate of the sampling widths. In the setting (ii) for the high dimensional case ($d\ge 2$), we also achieve the right convergence rate of the sampling $n$-widths for $1\le q \le 2 \le p \le \infty$ through a non-constructive method.