Given a sequence of Marcinkiewicz-Zygmund inequalities in $L_2$ on a usual compact space $\mathcal M$, Gr\"ochenig introduced the weighted least squares polynomials and the least squares quadrature from pointwise samples of a function, and obtained approximation theorems and quadrature errors. In this paper we obtain approximation theorems and quadrature errors on the sphere which are optimal. We also give upper bounds of the operator norms of the weighted least squares operators.

翻译：考虑到马克辛基维茨-齐格蒙德(Marcinkiewicz-Zygmund)在通常的紧凑空间上以2美元计算的一系列不平等情况,Gr\"chenig从一个函数的点样中引入了加权最小正方形多式和最小正方形二次方形,并获得了近似定理和二次差错。在本文中,我们在最理想的球体上获得了近似定理和二次差错。我们还给出了加权最小正方形操作员操作规范的上限 。