Optimal quadrature errors and sampling numbers for Sobolev spaces with logarithmic perturbation on spheres
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| Format: | Preprint |
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2024
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| _version_ | 1866910310373261312 |
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| author | Geng, Jiaxin Ling, Yun Li, Jiansong Wang, Heping |
| author_facet | Geng, Jiaxin Ling, Yun Li, Jiansong Wang, Heping |
| contents | In this paper, we study optimal quadrature errors, approximation numbers, and sampling numbers in $L_2(\Bbb S^d)$ for Sobolev spaces ${\rm H}^{α,β}(\Bbb S^d)$ with logarithmic perturbation on the unit sphere $\Bbb S^d$ in $\Bbb R^{d+1}$. First we obtain strong equivalences of the approximation numbers for ${\rm H}^{α,β}(\Bbb S^d)$ with $α>0$, which gives a clue to Open problem 3 as posed by Krieg and Vybíral in \cite{KV}. Second, for the optimal quadrature errors for ${\rm H}^{α,β}(\Bbb S^d)$, we use the "fooling" function technique to get lower bounds in the case $α>d/2$, and apply Hilbert space structure and Vybíral's theorem about Schur product theory to obtain lower bounds in the case $α=d/2,\,β>1/2$ of small smoothness, which confirms the conjecture as posed by Grabner and Stepanyukin in \cite{GS} and solves Open problem 2 in \cite{KV}. Finally, we employ the weighted least squares operators and the least squares quadrature rules to obtain approximation theorems and quadrature errors for ${\rm H}^{α,β}(\Bbb S^d)$ with $α>d/2$ or $α=d/2,\,β>1/2$, which are order optimal. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_16199 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimal quadrature errors and sampling numbers for Sobolev spaces with logarithmic perturbation on spheres Geng, Jiaxin Ling, Yun Li, Jiansong Wang, Heping Numerical Analysis 41A63, 65C05, 65D15, 65Y20 In this paper, we study optimal quadrature errors, approximation numbers, and sampling numbers in $L_2(\Bbb S^d)$ for Sobolev spaces ${\rm H}^{α,β}(\Bbb S^d)$ with logarithmic perturbation on the unit sphere $\Bbb S^d$ in $\Bbb R^{d+1}$. First we obtain strong equivalences of the approximation numbers for ${\rm H}^{α,β}(\Bbb S^d)$ with $α>0$, which gives a clue to Open problem 3 as posed by Krieg and Vybíral in \cite{KV}. Second, for the optimal quadrature errors for ${\rm H}^{α,β}(\Bbb S^d)$, we use the "fooling" function technique to get lower bounds in the case $α>d/2$, and apply Hilbert space structure and Vybíral's theorem about Schur product theory to obtain lower bounds in the case $α=d/2,\,β>1/2$ of small smoothness, which confirms the conjecture as posed by Grabner and Stepanyukin in \cite{GS} and solves Open problem 2 in \cite{KV}. Finally, we employ the weighted least squares operators and the least squares quadrature rules to obtain approximation theorems and quadrature errors for ${\rm H}^{α,β}(\Bbb S^d)$ with $α>d/2$ or $α=d/2,\,β>1/2$, which are order optimal. |
| title | Optimal quadrature errors and sampling numbers for Sobolev spaces with logarithmic perturbation on spheres |
| topic | Numerical Analysis 41A63, 65C05, 65D15, 65Y20 |
| url | https://arxiv.org/abs/2401.16199 |