High-dimensional sparse trigonometric approximation in the uniform norm and consequences for sampling recovery
<|reference_start|>High-dimensional sparse trigonometric approximation in the uniform norm and consequences for sampling recovery: 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].<|reference_end|>
@article{moeller2024high-dimensional,
    title={High-dimensional sparse trigonometric approximation in the uniform norm
  and consequences for sampling recovery},
    author={Moritz Moeller, Serhii Stasyuk, Tino Ullrich},
    journal={arXiv preprint arXiv:2407.15965},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.15965},
    primaryClass={math.NA cs.NA}
}