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 $q=p=2$.

中文翻译：

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配备高斯测度的 ℝd 上函数的最优数值积分和逼近

我们研究了 $\mathbb{R}^{d}$ 上积分的数值近似，配备了标准高斯测度 $\gamma $，对于属于高斯加权 Sobolev 空间 $W^{\alpha }_{p} 的被积函数(\mathbb{R}^{d}, \gamma )$ 的混合平滑度 $\alpha \in \mathbb{N}$ for $1 < p＜\infty $。我们证明了基于$n$个积分节点的最优求积收敛的渐近阶数，并提出了一种构造渐近最优求积的新方法。对于相关问题，我们通过类似的技术建立高斯加权空间 $L_{q}(\mathbb{R}^{d}, \gamma 中线性、柯尔莫哥洛夫和采样 $n$ 宽度的渐近阶$W^{\alpha }_{p}(\mathbb{R}^{d}, \gamma )$ 的单位球的 )$ 对于 $1 \leq q < p＜\infty $ 和 $q=p=2$。