Fast-Decaying Polynomial Reproduction

TL;DR

This paper introduces a novel framework for polynomial reproduction that allows basis functions to decay at infinity, enabling stable, convergent, and efficient approximation methods applicable to a wide range of schemes.

Contribution

It proposes the concept of fast decaying polynomial reproduction, extending traditional methods to non-compactly supported basis functions for improved stability and efficiency.

Findings

Stable and convergent approximation methods developed

Numerical verification of convergence rates and Lebesgue constants

Applicable to multivariate settings with various smoothness requirements

Abstract

Polynomial reproduction plays a relevant role in deriving error estimates for various approximation schemes. Local reproduction in a quasi-uniform setting is a significant factor in the estimation of error and the assessment of stability but for some computationally relevant schemes, such as Rescaled Localized Radial Basis Functions (RL-RBF), it becomes a limitation. To facilitate the study of a greater variety of approximation methods in a unified and efficient manner, this work proposes a framework based on fast decaying polynomial reproduction: we do not restrict to compactly supported basis functions, but we allow the basis function decay to infinity as a function of the separation distance. Implementing fast decaying polynomial reproduction provides stable and convergent methods, that can be smooth when approximating by moving least squares otherwise very efficient in the case of linear programming problems. All the results presented in this paper concerning the rate of convergence, the Lebesgue constant, the smoothness of the approximant, and the compactness of the support have been verified numerically, even in the multivariate setting.