Weighted least $\ell_p$ approximation on compact Riemannian manifolds
<|reference_start|>Weighted least $\ell_p$ approximation on compact Riemannian manifolds: Given a sequence of Marcinkiewicz-Zygmund inequalities in $L_2$ on a compact space, Gr\"ochenig 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,\tau}^r(\Bbb M),\, 0<\tau\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.<|reference_end|>
@article{li2024weighted,
    title={Weighted least $\ell_p$ approximation on compact Riemannian manifolds},
    author={Jiansong Li, Yun Ling, Jiaxin Geng, Heping Wang},
    journal={arXiv preprint arXiv:2402.19132},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.19132},
    primaryClass={math.NA cs.NA}
}