High-dimensional sparse trigonometric approximation in the uniform norm and consequences for sampling recovery

TL;DR

This paper advances high-dimensional sparse trigonometric approximation by establishing new bounds that control the influence of dimension, with implications for sampling recovery in Besov classes and extending classical inequalities.

Contribution

It provides new approximation guarantees for high-dimensional trigonometric approximation, refining classical inequalities and analyzing the dimension's impact on sampling recovery.

Findings

Approximation error scales at most quadratically with inverse accuracy

Dimension influence is limited to a actor and a logarithmic term

Results have implications for sampling recovery in Besov classes

Abstract

Recent findings by Jahn, T. Ullrich, Voigtlaender [10] relate non-linear sampling numbers for the square norm to quantities involving trigonometric best $m-$term approximation errors in the uniform norm. Here we establish new results for sparse trigonometric approximation with respect to the high-dimensional setting, where the influence of the dimension $d$ has to be controlled. In particular, we focus on best $m-$term trigonometric approximation for (unweighted) Wiener classes in $L_q$ and give precise constants. Our main results are approximation guarantees where the number of terms $m$ scales at most quadratic in the inverse accuracy $1/\varepsilon$. Providing a refined version of the classical Nikol'skij inequality we are able to extrapolate the $L_q$-result to $L_\infty$ while limiting the influence of the dimension to a $\sqrt{d}$-factor and an additonal $\log$-term in the size of the (rectangular) spectrum. This has consequences for the tractable sampling recovery via $\ell_1$-minimization of functions belonging to certain Besov classes with bounded mixed smoothness. This complements polynomial tractability results recently given by Krieg [12].