Optimal numerical integration for functions in fractional Gaussian Sobolev spaces
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910141637459968 |
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| author | Nguyen, Van Kien |
| author_facet | Nguyen, Van Kien |
| contents | This paper investigates the numerical approximation of integrals for functions in fractional Gaussian Sobolev spaces $W^s_{p}(\mathbb{R}^d,γ)$ 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 $W^s_{2}(\mathbb{R}^d,γ)$ coincide with Hermite spaces $\mathcal{H}^s(\mathbb{R}^d,γ)$ 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 $W^s_{p,G}(\mathbb{R}^d,γ)$ defined via the Gagliardo seminorm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_03659 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Optimal numerical integration for functions in fractional Gaussian Sobolev spaces Nguyen, Van Kien Numerical Analysis This paper investigates the numerical approximation of integrals for functions in fractional Gaussian Sobolev spaces $W^s_{p}(\mathbb{R}^d,γ)$ 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 $W^s_{2}(\mathbb{R}^d,γ)$ coincide with Hermite spaces $\mathcal{H}^s(\mathbb{R}^d,γ)$ 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 $W^s_{p,G}(\mathbb{R}^d,γ)$ defined via the Gagliardo seminorm. |
| title | Optimal numerical integration for functions in fractional Gaussian Sobolev spaces |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2604.03659 |