Weighted least $\ell_p$ approximation on compact Riemannian manifolds
Jiansong Li, Yun Ling, Jiaxin Geng, Heping Wang
TL;DR
This work extends the weighted least $\ell_p$ approximation framework from $L_2$ to general $L_p$ norms on compact Riemannian manifolds by leveraging $L_p$-Marcinkiewicz–Zygmund families. It establishes error bounds for approximations in $L_q$ and for quadrature across Sobolev $H_p^r(\Bbb M)$ and Besov $B_{p,\tau}^r(\Bbb M)$ spaces with $r>d/p$, yielding rates $\|f-L_{n,p}^{\Bbb M}(f)\|_{L_q(\Bbb M)} \lesssim n^{-r+d(1/p-1/q)_+}$ (and analogous for $p=\infty$) and quadrature errors $\lesssim n^{-r}$. The results are shown to be asymptotically optimal via sharp sampling-number and optimal-quadrature estimates, and they apply to classical manifolds such as the torus and sphere. The work broadens the applicability of manifold-based polynomial approximation by providing a coherent, provably optimal sampling-based framework for Sobolev and Besov spaces.
Abstract
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.