Widths of embeddings of Gaussian Sobolev spaces

Abstract

In this paper, we investigate the approximation problem for functions in Gaussian Sobolev spaces $W^s_p(\mathbb{R}^d, γ)$ of smoothness $s > 0$, where the approximation error is measured in the Gaussian Lebesgue space $L_q(\mathbb{R}^d, γ)$. 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.