Optimal numerical integration and approximation of functions on ℝ^d equipped with Gaussian measure

07/04/2022

∙

by   Dinh Dũng, et al.

∙

0

∙

share

We investigate the numerical approximation of integrals over ℝ^d equipped with the standard Gaussian measure γ for integrands belonging to the Gaussian-weighted Sobolev spaces W^α_p(ℝ^d, γ) of mixed smoothness α∈ℕ for 1 < p < ∞. We prove the asymptotic order of convergence rate of optimal quadratures based on n integration nodes and propose a novel method for constructing asymptotically optimal quadratures. As on related problems, we establish by a similar technique the asymptotic order of the linear, Kolmogorov and sampling n-widths in Gaussian-weighted space L_q(ℝ^d, γ) of the unit ball of W^α_p(ℝ^d, γ) for 1 ≤ q < p < ∞ and q=p=2.