论文标题
$ \ mathbb {r}^d $配备高斯度量的最佳数值集成和函数的近似
Optimal numerical integration and approximation of functions on $\mathbb{R}^d$ equipped with Gaussian measure
论文作者
论文摘要
我们调查了$ \ mathbb {r}^d $在$ \ mathbb {r}^d $上的分析近似值，配备了标准的高斯度量$γ$属于Gaussian-Weighted sobolev sobolev sobolev sobolev sobolev sobolev sobolev space $ w^α_p（\ \ mthbb {r} \ infty $。我们证明了基于$ n $ Integration节点的最佳四二次交汇的渐近顺序，并提出了一种构造渐近最佳四倍体的新方法。至于相关问题，我们通过类似的技术建立线性，kolmogorov的渐近顺序和在高斯加权空间$ l_q（\ Mathbb {r}^d，γ）$ $ w^α_p（\ mathbbbb {r Mathbb {r} pe <per）中的$ n $ widths $ l_q（\ mathbb {r}^d，γ）$ \ infty $和$ q = p = 2 $。
We investigate the numerical approximation of integrals over $\mathbb{R}^d$ equipped with the standard Gaussian measure $γ$ for integrands belonging to the Gaussian-weighted Sobolev spaces $W^α_p(\mathbb{R}^d, γ)$ of mixed smoothness $α\in \mathbb{N}$ for $1 < p < \infty$. We prove the asymptotic order of the convergence of optimal quadratures based on $n$ integration nodes and propose a novel method for constructing asymptotically optimal quadratures. As for related problems, we establish by a similar technique the asymptotic order of the linear, Kolmogorov and sampling $n$-widths in the Gaussian-weighted space $L_q(\mathbb{R}^d, γ)$ of the unit ball of $W^α_p(\mathbb{R}^d, γ)$ for $1 \leq q < p < \infty$ and $q=p=2$.
