How anisotropic mixed smoothness affects the decay of singular numbers of Sobolev embeddings

TL;DR

This paper investigates how anisotropic mixed smoothness influences the decay of singular numbers in Sobolev embeddings, providing precise asymptotic constants and improved bounds for high-dimensional approximation errors.

Contribution

It offers the first explicit asymptotic constants and refined preasymptotic bounds for singular numbers in tensor product Sobolev embeddings with anisotropic smoothness.

Findings

Derived the correct asymptotic constant for singular number decay.

Provided explicit preasymptotic bounds improving existing results.

Refined error bounds for moderately increasing smoothness vectors.

Abstract

We continue the research on the asymptotic and preasymptotic decay of singular numbers for tensor product Hilbert-Sobolev type embeddings in high dimensions with special emphasis on the influence of the underlying dimension $d$. The main focus in this paper lies on tensor products involving univariate Sobolev type spaces with different smoothness. We study the embeddings into $L_2$ and $H^1$. In other words, we investigate the worst-case approximation error measured in $L_2$ and $H^1$ when only $n$ linear samples of the function are available. Recent progress in the field shows that accurate bounds on the singular numbers are essential for recovery bounds using only function values. The asymptotic bounds in our setting are known for a long time. In this paper we contribute the correct asymptotic constant and explicit bounds in the preasymptotic range for $n$. We complement and improve on several results in the literature. In addition, we refine the error bounds coming from the setting where the smoothness vector is moderately increasing, which has been already studied by Papageorgiou and Wo{\'z}niakowski.