Weighted Least \(\ell _p\) Approximation on Compact Riemannian Manifolds

Given a sequence of Marcinkiewicz–Zygmund inequalities in \(L_2\) $L_2$ on a compact space, Gröchenig (J Approx Theory 257:105455, 2020) discussed weighted least squares approximation and least squares quadrature. Inspired by this work, for all \(1\le p\le \infty \) $1\le p\le \infty$ , we develop weighted least \(\ell _p\) ${\ell }_p$ approximation induced by a sequence of Marcinkiewicz–Zygmund inequalities in \(L_p\) $L_p$ on a compact smooth Riemannian manifold \(\mathbb M\) $\mathbb{M}$ 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 \(L_q,\,1\le q\le \infty \) $L_q,1\le q\le \infty$ , and least quadrature errors for both Sobolev spaces \(H_p^r(\mathbb M), \, r>d/p\) $H_p^r(\mathbb{M}),r>d/p$ generated by eigenfunctions associated with the Laplace-Beltrami operator and Besov spaces \(B_{p,\gamma }^r(\mathbb M),\, 0<\gamma \le \infty ,\, r>d/p \) $B_{p,\gamma }^r(\mathbb{M}),0<\gamma \le \infty ,r>d/p$ 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.