Optimal sampling recovery of mixed order Sobolev embeddings via discrete Littlewood-Paley type characterizations

TL;DR

This paper analyzes the optimality of sampling recovery algorithms for multivariate functions with mixed smoothness, providing sharp bounds and characterizing their asymptotic behavior using discrete Littlewood-Paley techniques.

Contribution

It introduces a systematic analysis of Smolyak algorithms in Besov-Lizorkin-Triebel spaces, establishing sharp bounds for sampling widths and closing gaps in existing literature.

Findings

Sharp upper bounds for sampling widths in various cases.

Linear Smolyak interpolation is optimal in certain regimes.

Asymptotic behavior matches Gelfand n-widths in key cases.

Abstract

In this paper we consider the $L_q$-approximation of multivariate periodic functions $f$ with $L_p$-bounded mixed derivative (difference). The (possibly non-linear) reconstruction algorithm is supposed to recover the function from function values, sampled on a discrete set of $n$ sampling nodes. The general performance is measured in terms of (non-)linear sampling widths $\varrho_n$. We conduct a systematic analysis of Smolyak type interpolation algorithms in the framework of Besov-Lizorkin-Triebel spaces of dominating mixed smoothness based on specifically tailored discrete Littlewood-Paley type characterizations. As a consequence, we provide sharp upper bounds for the asymptotic order of the (non-)linear sampling widths in various situations and close some gaps in the existing literature. For example, in case $2\leq p<q<\infty$ and $r>1/p$ the linear sampling widths $\varrho_n^{\text{lin}}(S^r_pW(\mathbb{T}^d),L_q(\mathbb{T}^d))$ and $\varrho^{\text{lin}}_n(S^r_{p,\infty}B(\mathbb{T}^d),L_q(\mathbb{T}^d))$ show the asymptotic behavior of the corresponding Gelfand $n$-widths, whereas in case $1 < p < q \leq 2$ and $r>1/p$ the linear sampling widths match the corresponding linear widths. In the mentioned cases linear Smolyak interpolation based on univariate classical trigonometric interpolation turns out to be optimal.