Given a sequence of Marcinkiewicz-Zygmund inequalities in $L_2$ 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 $\ell_p$ 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,\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.

翻译：摘要：基于紧致空间上$L_2$中的Marcinkiewicz-Zygmund不等式序列，Gröchenig在文献\cite{G}中探讨了加权最小二乘逼近及最小二乘求积问题。受此启发，本文针对所有$1\le p\le\infty$情形，在带有归一化黎曼测度的紧致光滑黎曼流形$\Bbb M$（典型示例包括环面与球面）上，发展由$L_p$中Marcinkiewicz-Zygmund不等式序列诱导的加权最小$\ell_p$逼近理论。我们建立了相应的逼近定理，误差分别以$L_q\,(1\le q\le\infty)$度量，并给出了对于两类函数空间的求积误差：一类是由Laplace-Beltrami算子特征函数生成的Sobolev空间$H_p^r(\Bbb M)\, (r>d/p)$，另一类是基于最佳多项式逼近定义的Besov空间$B_{p,\tau}^r(\Bbb M)\, (0<\tau\le \infty,\, r>d/p)$。最后，通过给出上述空间的采样数精确估计与最优求积误差界，论证了所得结果的最优性。