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

TL;DR

This paper studies the asymptotic behavior of optimal numerical integration methods over Gaussian spaces, introducing new quadrature constructions and analyzing widths in Gaussian-weighted Sobolev spaces.

Contribution

It provides the asymptotic order of convergence for optimal quadratures and proposes a novel method for their construction in Gaussian-weighted Sobolev spaces.

Findings

Asymptotic order of convergence for optimal quadratures established.

New method for constructing asymptotically optimal quadratures proposed.

Analysis of widths in Gaussian-weighted spaces for various norms.

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 $W^\alpha_p(\mathbb{R}^d, \gamma)$ 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 $q=p=2$.