Direct and Inverse Results on Bounded Domains for Meshless Methods via Localized Bases on Manifolds

TL;DR

This paper establishes direct and inverse approximation estimates for localized kernel-based spaces on manifolds, enabling efficient meshless methods for surface approximation and analysis.

Contribution

It introduces a general framework for localized Lagrange functions on manifolds, extending previous constructions to Sobolev-Matérn kernels and providing computationally efficient approximation tools.

Findings

Derived direct and inverse estimates for kernel approximation spaces.

Constructed localized Lagrange functions applicable to Sobolev-Matérn kernels.

Generalized previous surface spline constructions to broader manifold settings.

Abstract

This article develops direct and inverse estimates for certain finite dimensional spaces arising in kernel approximation. Both the direct and inverse estimates are based on approximation spaces spanned by local Lagrange functions which are spatially highly localized. The construction of such functions is computationally efficient and generalizes the construction given by the authors for restricted surface splines on $\mathbb{R}^d$. The kernels for which the theory applies includes the Sobolev-Mat\'ern kernels for closed, compact, connected, $C^\infty$ Riemannian manifolds.