Weighted least $\ell_p$ approximation on compact Riemannian manifolds

Given a sequence of Marcinkiewicz-Zygmund inequalities in $L2$ 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 $\ellp$ approximation induced by a sequence of Marcinkiewicz-Zygmund inequalities in $Lp$ 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 $Lq,\,1\le q\le\infty$, and least quadrature errors for both Sobolev spaces $Hp^r(\Bbb M), \, r>d/p$ 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.