Weighted least \(\ell_p\) approximation on compact Riemannian manifolds (Q6632433)

In this research article, the authors develop weighted least \(l_p\) approximation induced by a sequence of Marcinkiewicz-Zygmund inequalities in \(L_p\) on a compact smooth Riemannian manifold \(\mathbb{M}\) with normailzed Riemannian measure for all \(1 \le p \le \infty\). They derive corresponding approximation theorems with the error measured in \(L_q\), \(1 \le q \le \infty\), and least quadrature errors for both Sobolev spaces \(H_p^r(\mathbb{M})\), \(r > \frac{d}{p}\) generated by eigenfunctions associated with the Laplace-Beltrami operator and Besov spaces \(B_{p,y}^r(\mathbb{M})\), \(0 \le y \le \infty\), \(r >\frac{d}{p}\) defined by best ``polynomial'' approximation. In the end, they discuss the optimality of the obtained results by giving sharp estimates of sampling numbers and optimal quadrature errors for the aforementioned spaces.