Weighted least $\ell_p$ approximation on compact Riemannian manifolds
Abstract
Given a sequence of MarcinkiewiczZygmund 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 MarcinkiewiczZygmund 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 LaplaceBeltrami 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2024
 DOI:
 10.48550/arXiv.2402.19132
 arXiv:
 arXiv:2402.19132
 Bibcode:
 2024arXiv240219132L
 Keywords:

 Mathematics  Numerical Analysis;
 41A17;
 41A55;
 41A63;
 65D15;
 65D30;
 65D32
 EPrint:
 23 pages