Optimal numerical integration and approximation of functions on $\mathbb{R}^d$ equipped with Gaussian measure

Abstract: We investigate the numerical approximation of integrals over $\mathbb{R}<sup>d$ 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 $1 &lt; p &lt; \infty$. 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 $1 \leq q &lt; p &lt; \infty$ and $q=p=2$.