Widths of embeddings of Gaussian Sobolev spaces

TL;DR

This paper analyzes the asymptotic behavior of various widths for Gaussian Sobolev spaces, revealing how approximation complexity varies with parameters and contributing to high-dimensional approximation theory.

Contribution

It provides the exact asymptotic order of Kolmogorov, linear, and sampling widths for Gaussian Sobolev spaces across different parameter regimes.

Findings

Exact asymptotic order of widths determined for different regimes

Reveals distinct approximation phenomena for different parameter settings

Advances understanding of high-dimensional Gaussian measure approximation

Abstract

In this paper, we investigate the approximation problem for functions in Gaussian Sobolev spaces $W^s_p(\mathbb{R}^d, \gamma)$ of smoothness $s > 0$, where the approximation error is measured in the Gaussian Lebesgue space $L_q(\mathbb{R}^d, \gamma)$. Such function spaces naturally arise in the analysis of high-dimensional problems with Gaussian measures and play an important role in various applications, including uncertainty quantification and stochastic modeling. Our main objective is to analyze the asymptotic behavior of fundamental quantities that characterize the complexity of the approximation problem. In particular, we determine the exact asymptotic order of several classes of widths, including Kolmogorov, linear, and sampling widths, which quantify the optimal performance of different approximation methods. The obtained results cover the parameter regimes $1 \leq q < p < \infty$ and $p = q = 2$, where distinct phenomena in terms of approximation rates can be observed.