Optimal numerical integration for functions in fractional Gaussian Sobolev spaces

Abstract

This paper investigates the numerical approximation of integrals for functions in fractional Gaussian Sobolev spaces $W_p^s({\mathbb{R}}^d,{\hat{I}}^3)$ with dominating mixed smoothness defined via kernel related to the fractional Ornstein-Uhlenbeck operator. Building upon quadrature rules for fractional Sobolev spaces on the unit cube $[-\frac{1}{2},\frac{1}{2}{]}^d$, we construct quadrature schemes on ${\mathbb{R}}^d$ that achieve the same rate of convergence. As a consequence, we establish the optimal asymptotic order of the integration error in the regime $1<p<\infty$ and $s>\frac{1}{p}$, $s\notin \mathbb{N}$. Furthermore, we show that the fractional Gaussian Sobolev spaces $W_2^s({\mathbb{R}}^d,{\hat{I}}^3)$ coincide with Hermite spaces ${\mathcal{H}}^s({\mathbb{R}}^d,{\hat{I}}^3)$ characterized by the weighted ${\ell }_2$-summability of their Fourier-Hermite coefficients. From this, we derive the optimal asymptotic order of the integration error for functions in these spaces for all $s>\frac{1}{2}$. We also establish the corresponding optimal asymptotic order for functions in fractional Gaussian Sobolev spaces $W_{p,G}^s({\mathbb{R}}^d,{\hat{I}}^3)$ defined via the Gagliardo seminorm.

20 pages