Weighted least ₚ approximation on compact Riemannian manifolds

Given a sequence of Marcinkiewicz-Zygmund inequalities in L₂ on a compact space, Gr\"ochenig in G discussed weighted least squares approximation and least squares quadrature. Inspired by this work, for all 1 p, we develop weighted least ₚ approximation induced by a sequence of Marcinkiewicz-Zygmund inequalities in Lₚ on a compact smooth Riemannian manifold 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 q, and least quadrature errors for both Sobolev spaces Hₚʳ (M), \, r>d/p generated by eigenfunctions associated with the Laplace-Beltrami operator and Besov spaces B₏, ʳ (M), \, 0d/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.