We present a general framework, treating Lipschitz domains in Riemannian manifolds, that provides conditions guaranteeing the existence of norming sets and generalized local polynomial reproduction - a powerful tool used in the analysis of various mesh-free methods and a mesh-free method in its own right. As a key application, we prove the existence of smooth local polynomial reproductions on compact subsets of algebraic manifolds in $\mathbb{R}^n$ with Lipschitz boundary. These results are then applied to derive new findings on the existence, stability, regularity, locality, and approximation properties of shape functions for a coordinate-free moving least squares approximation method on algebraic manifolds, which operates directly on point clouds without requiring tangent plane approximations.

翻译：我们提出了一个处理黎曼流形中Lipschitz域的一般框架，该框架提供了保证范数集存在性和广义局部多项式再生存在的条件——这是分析各种无网格方法时使用的强大工具，其本身也是一种无网格方法。作为一个关键应用，我们证明了在具有Lipschitz边界的$\mathbb{R}^n$代数流形紧子集上光滑局部多项式再生的存在性。这些结果随后被应用于推导关于代数流形上坐标无关移动最小二乘逼近方法（该方法直接在点云上操作而无需切平面逼近）的形函数存在性、稳定性、正则性、局部性及逼近性质的新结论。