Weighted least $\ell_p$ approximation on compact Riemannian manifolds
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866911786404413440 |
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| author | Li, Jiansong Ling, Yun Geng, Jiaxin Wang, Heping |
| author_facet | Li, Jiansong Ling, Yun Geng, Jiaxin Wang, Heping |
| contents | Given a sequence of Marcinkiewicz-Zygmund inequalities in $L_2$ on a compact space, Gröchenig in \cite{G} discussed weighted least squares approximation and least squares quadrature. Inspired by this work, for all $1\le p\le\infty$, we develop weighted least $\ell_p$ approximation induced by a sequence of Marcinkiewicz-Zygmund inequalities in $L_p$ on a compact smooth Riemannian manifold $\Bbb M$ with normalized Riemannian measure (typical examples are the torus and the sphere). In this paper we derive corresponding approximation theorems with the error measured in $L_q,\,1\le q\le\infty$, and least quadrature errors for both Sobolev spaces $H_p^r(\Bbb M), \, r>d/p$ generated by eigenfunctions associated with the Laplace-Beltrami operator and Besov spaces $B_{p,τ}^r(\Bbb M),\, 0<τ\le \infty, r>d/p $ defined by best polynomial approximation. Finally, we discuss the optimality of the obtained results by giving sharp estimates of sampling numbers and optimal quadrature errors for the aforementioned spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_19132 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weighted least $\ell_p$ approximation on compact Riemannian manifolds Li, Jiansong Ling, Yun Geng, Jiaxin Wang, Heping Numerical Analysis 41A17, 41A55, 41A63, 65D15, 65D30, 65D32 Given a sequence of Marcinkiewicz-Zygmund inequalities in $L_2$ on a compact space, Gröchenig in \cite{G} discussed weighted least squares approximation and least squares quadrature. Inspired by this work, for all $1\le p\le\infty$, we develop weighted least $\ell_p$ approximation induced by a sequence of Marcinkiewicz-Zygmund inequalities in $L_p$ on a compact smooth Riemannian manifold $\Bbb M$ with normalized Riemannian measure (typical examples are the torus and the sphere). In this paper we derive corresponding approximation theorems with the error measured in $L_q,\,1\le q\le\infty$, and least quadrature errors for both Sobolev spaces $H_p^r(\Bbb M), \, r>d/p$ generated by eigenfunctions associated with the Laplace-Beltrami operator and Besov spaces $B_{p,τ}^r(\Bbb M),\, 0<τ\le \infty, r>d/p $ defined by best polynomial approximation. Finally, we discuss the optimality of the obtained results by giving sharp estimates of sampling numbers and optimal quadrature errors for the aforementioned spaces. |
| title | Weighted least $\ell_p$ approximation on compact Riemannian manifolds |
| topic | Numerical Analysis 41A17, 41A55, 41A63, 65D15, 65D30, 65D32 |
| url | https://arxiv.org/abs/2402.19132 |