Generalized local polynomial reproductions

TL;DR

This paper develops a general framework for local polynomial reproduction on Lipschitz domains in Riemannian manifolds, enabling mesh-free approximation methods directly on point clouds in algebraic manifolds.

Contribution

It introduces a unified approach to local polynomial reproduction on Lipschitz domains, with applications to mesh-free methods on algebraic manifolds, including stability and approximation properties.

Findings

Existence of smooth local polynomial reproductions on algebraic manifolds.

New stability and regularity results for shape functions in mesh-free approximation.

Framework applicable to coordinate-free moving least squares methods on point clouds.

Abstract

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. 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.