Generalized Local Polynomial Reproductions

We present a general framework, treating Lipschitz domains in Riemannian manifolds, that provides conditions guaranteeing the existence of norming sets and generalized local polynomial reproductions—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 \(\mathbb {R}^n\) ${\mathbb{R}}^n$ 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. There are two appendices: the first derives high order Markov inequalities for polynomials on algebraic manifolds and the second gives instructions for calculating the dimension of the space of degree m polynomials restricted to a real algebraic variety.