Pseudo $s$-Numbers of Embeddings of Gaussian Weighted Sobolev Spaces

Abstract

In this paper, we study the approximation problem for functions in the Gaussian-weighted Sobolev space $W_p^{\alpha }({\mathbb{R}}^d,\gamma )$ of mixed smoothness $\alpha \in \mathbb{N}$ with error measured in the Gaussian-weighted space $L_q({\mathbb{R}}^d,\gamma )$. We obtain the exact asymptotic order of pseudo $s$-numbers for the cases $1\le q<p<\infty$ and $p=q=2$. Additionally, we also obtain an upper bound and a lower bound for pseudo $s$-numbers of the embedding of $W_2^{\alpha }({\mathbb{R}}^d,\gamma )$ into $L_{\infty }^{\sqrt{g}}({\mathbb{R}}^d)$. Our result is an extension of that obtained in Dinh Dũng and Van Kien Nguyen (IMA Journal of Numerical Analysis, 2023) for approximation and Kolmogorov numbers.

17 pages