Generalized local polynomial reproductions

Thomas Hangelbroek; Christian Rieger; Grady B. Wright

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arXiv · 2410.12973

Generalized local polynomial reproductions

Thomas Hangelbroek·Christian Rieger·Grady B. Wright

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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.

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BibTeX RIS

Thomas Hangelbroek, Christian Rieger, Grady B. Wright. 2024-10-16. Generalized local polynomial reproductions. https://arxiv.org/abs/2410.12973

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