Optimal numerical integration for functions in fractional Gaussian Sobolev spaces

TL;DR

This paper develops optimal quadrature schemes for integrating functions in fractional Gaussian Sobolev spaces, achieving the best possible convergence rates and connecting these spaces with Hermite spaces.

Contribution

It constructs quadrature rules on  that match the convergence rates known on the unit cube and establishes the equivalence of fractional Gaussian Sobolev spaces with Hermite spaces.

Findings

Achieves optimal asymptotic order of integration error for fractional Gaussian Sobolev spaces.

Constructs quadrature schemes on  with the same convergence rate as on the unit cube.

Shows the equivalence of fractional Gaussian Sobolev spaces with Hermite spaces for certain parameters.

Abstract

This paper investigates the numerical approximation of integrals for functions in fractional Gaussian Sobolev spaces $W^s_{p}(\mathbb{R}^d,\gamma)$ 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 $[-\tfrac{1}{2}, \tfrac{1}{2}]^d$, 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 $s > \frac{1}{p}$, $s\not \in \mathbb{N}$. Furthermore, we show that the fractional Gaussian Sobolev spaces $W^s_{2}(\mathbb{R}^d,\gamma)$ coincide with Hermite spaces $\mathcal{H}^s(\mathbb{R}^d,\gamma)$ 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 $W^s_{p,G}(\mathbb{R}^d,\gamma)$ defined via the Gagliardo seminorm.