We investigate the numerical approximation of integrals over $\mathbb{R}^d$ equipped with the standard Gaussian measure $\gamma$ for integrands belonging to the Gaussian-weighted Sobolev spaces $W^\alpha_p(\mathbb{R}^d, \gamma)$ of mixed smoothness $\alpha \in \mathbb{N}$ 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 $L_q(\mathbb{R}^d, \gamma)$ of the unit ball of $W^\alpha_p(\mathbb{R}^d, \gamma)$ for $1 \leq q < p < \infty$ and $q=p=2$.

翻译：本文研究在标准高斯测度 $\gamma$ 下，对定义于 $\mathbb{R}^d$ 上且属于高斯加权混合光滑性 Sobolev 空间 $W^\alpha_p(\mathbb{R}^d, \gamma)$（其中光滑性指标 $\alpha \in \mathbb{N}$，$1 < p < \infty$）的被积函数进行积分数值逼近的问题。我们证明了基于 $n$ 个积分节点的最优求积公式的渐近收敛阶，并提出了一种构造渐近最优求积公式的新方法。针对相关问题，我们采用类似技术，建立了当 $1 \leq q < p < \infty$ 及 $q=p=2$ 时，高斯加权空间 $L_q(\mathbb{R}^d, \gamma)$ 中 $W^\alpha_p(\mathbb{R}^d, \gamma)$ 单位球的线性宽度、Kolmogorov 宽度和采样 $n$ 宽度的渐近阶。