TY - JOUR
AU - Kien Nguyen, Van
AB - 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 $q=p=2$.
TI - Optimal numerical integration and approximation of functions on ℝd equipped with Gaussian measure
JO - IMA Journal of Numerical Analysis
DO - 10.1093/imanum/drad051
DA - 2023-08-02
UR - https://www.deepdyve.com/lp/oxford-university-press/optimal-numerical-integration-and-approximation-of-functions-on-d-Frtv11ezg0
SP - 1242
EP - 1267
VL - 44
IS - 2
DP - DeepDyve
ER -