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

Abstract

We investigate the numerical approximation of integrals over ${\mathbb{R}}^d$ equipped with the standard Gaussian measure $\gamma$ for integrands belonging to the Gaussian-weighted Sobolev spaces $W_p^{\alpha }({\mathbb{R}}^d,\gamma )$ 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 $W_p^{\alpha }({\mathbb{R}}^d,\gamma )$ for $1\le q<p<\infty$ and $q=p=2$.

arXiv comment23 pages