The recovery of multivariate functions and estimating their integrals from finitely many samples is one of the central tasks in modern approximation theory. Marcinkiewicz--Zygmund inequalities provide answers to both the recovery and the quadrature aspect. In this paper, we put ourselves on the $q$-dimensional sphere $\mathbb{S}^q$, and investigate how well continuous $L_p$-norms of polynomials $f$ of maximum degree $n$ on the sphere $\mathbb{S}^q$ can be discretized by positively weighted $L_p$-sum of finitely many samples, and discuss the relationship between the offset between the continuous and discrete quantities, the number and distribution of the (deterministic or randomly chosen) sample points $\xi_1,\ldots,\xi_N$ on $\mathbb{S}^q$, the dimension $q$, and the polynomial degree $n$.

翻译：多元函数的恢复以及从有限样本中估计其积分是现代逼近理论的核心问题之一。Marcinkiewicz--Zygmund不等式同时为函数恢复与求积问题提供了理论解答。本文以$q$维球面$\mathbb{S}^q$为研究空间，探讨如何通过带正权的$L_p$求和将球面$\mathbb{S}^q$上最高次数为$n$的多项式$f$的连续$L_p$范数进行离散化，并分析连续量与离散量之间的偏移量、球面$\mathbb{S}^q$上（确定性或随机选取的）样本点$\xi_1,\ldots,\xi_N$的数量与分布、维度$q$及多项式次数$n$之间的关联。