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

TL;DR

This paper determines the precise asymptotic behavior of pseudo s-numbers for embeddings of Gaussian-weighted Sobolev spaces, extending previous results to new parameter ranges and providing bounds for specific embeddings.

Contribution

It provides exact asymptotic orders of pseudo s-numbers for Gaussian-weighted Sobolev space embeddings, extending prior work to additional cases and bounds.

Findings

Exact asymptotic order of pseudo s-numbers for certain embeddings.

Upper and lower bounds for embeddings into weighted L-infinity spaces.

Extension of previous results on approximation and Kolmogorov numbers.

Abstract

In this paper, we study the approximation problem for functions in the Gaussian-weighted Sobolev space $W^\alpha_p(\mathbb{R}^d, \gamma)$ 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 $1 \leq q< p < \infty$ and $p=q=2$. Additionally, we also obtain an upper bound and a lower bound for pseudo $s$-numbers of the embedding of $W^\alpha_2(\mathbb{R}^d, \gamma)$ into $L_{\infty}^{\sqrt{g}}(\mathbb{R}^d)$. Our result is an extension of that obtained in Dinh D\~ung and Van Kien Nguyen (IMA Journal of Numerical Analysis, 2023) for approximation and Kolmogorov numbers.