High-dimensional sparse trigonometric approximation in the uniform norm and consequences for sampling recovery
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arXiv
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| Format: | Preprint |
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2024
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| author | Moeller, Moritz Stasyuk, Serhii Ullrich, Tino |
| author_facet | Moeller, Moritz Stasyuk, Serhii Ullrich, Tino |
| contents | 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]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_15965 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | High-dimensional sparse trigonometric approximation in the uniform norm and consequences for sampling recovery Moeller, Moritz Stasyuk, Serhii Ullrich, Tino Numerical Analysis 94A20, 41A46, 46E15, 42B35, 41A25, 65D15, 41A6 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]. |
| title | High-dimensional sparse trigonometric approximation in the uniform norm and consequences for sampling recovery |
| topic | Numerical Analysis 94A20, 41A46, 46E15, 42B35, 41A25, 65D15, 41A6 |
| url | https://arxiv.org/abs/2407.15965 |