Weighted sampling recovery of functions with mixed smoothness
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912694298214400 |
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| author | Dũng, Dinh |
| author_facet | Dũng, Dinh |
| contents | We studied linear weighted sampling algorithms and their optimality for approximate recovery of functions with mixed smoothness on $\mathbb{R}^d$ from a set of $n$ their sampled values. Functions to be recovered are in weighted Sobolev spaces $W^r_{p,w}(\mathbb{R}^d)$ of mixed smoothness, and the approximation error is measured by the norm of the weighted Lebesgue space $L_{q,w}(\mathbb{R}^d)$. Here, the weight $w$ is a tensor-product Freud-type weight. The optimality of linear sampling algorithms is investigated in terms of sampling $n$-widths. We constructed linear sampling algorithms on sparse grids of sampled points which form a step hyperbolic cross in the function domain, and which give upper bounds for the corresponding sampling $n$-widths. We proved that in the one-dimensional case, these algorithms realize the exact convergence rate of the $n$-sampling widths. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_16400 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weighted sampling recovery of functions with mixed smoothness Dũng, Dinh Numerical Analysis We studied linear weighted sampling algorithms and their optimality for approximate recovery of functions with mixed smoothness on $\mathbb{R}^d$ from a set of $n$ their sampled values. Functions to be recovered are in weighted Sobolev spaces $W^r_{p,w}(\mathbb{R}^d)$ of mixed smoothness, and the approximation error is measured by the norm of the weighted Lebesgue space $L_{q,w}(\mathbb{R}^d)$. Here, the weight $w$ is a tensor-product Freud-type weight. The optimality of linear sampling algorithms is investigated in terms of sampling $n$-widths. We constructed linear sampling algorithms on sparse grids of sampled points which form a step hyperbolic cross in the function domain, and which give upper bounds for the corresponding sampling $n$-widths. We proved that in the one-dimensional case, these algorithms realize the exact convergence rate of the $n$-sampling widths. |
| title | Weighted sampling recovery of functions with mixed smoothness |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2405.16400 |