Fast-Decaying Polynomial Reproduction

TL;DR

This work proposes a framework based on fast decaying polynomial reproduction, which does not restrict to compactly supported basis functions, but allows the basis function decay to infinity as a function of the separation distance.

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.

Figures (10)

• Figure 1: The weight function $w(x)=e^{-x^2} \in \mathcal{C}^{\infty}(\mathbb{R})$, the nodes are $5$ uniformly perturbed equispaced nodes in $[-1,1]$. From left to right the basis functions reproduce the polynomials of degrees $m=0, 3$, and $4$ respectively. In this numerical test $\delta = 5h_{X,\Omega}$.
• Figure 2: Basis functions of Theorem \ref{['thm_pr_fast_decay_exp']}. The weight function coincides with $e^{-x} \in \mathcal{C}(\mathbb{R})$ and the approxiomation nodes are $5$ uniformly perturbed equispaced nodes in $[-1,1]$. From left to right the basis functions reproduce the polynomials of degrees $0, 3$ and $4$ respectively. In this numerical test $\delta = 5h_{X,\Omega}$.
• Figure 3: Basis functions of equation \ref{['approximant_linear_problem_optimization_form']}. The weight function coincides with $e^{-x^2} \in \mathcal{C}^{\infty}(\mathbb{R})$ and the approxiomation nodes are $5$ uniformly perturbed equispaced nodes in $[-1,1]$. From left to right the basis functions reproduce the polynomials of degree $0, 3$ and $4$ respectively. In this numerical test $\delta = 5h_{X,\Omega}$.
• Figure 4: Basis functions of equation \ref{['approximant_linear_problem_optimization_form']}. The weight function coincides with $e^{-x} \in \mathcal{C}(\mathbb{R})$ and the approxiomation nodes are $5$ uniformly perturbed equispaced nodes in $[-1,1]$. From left to right the basis functions reproduce the polynomials of degree $0, 3$ and $4$ respectively. In this numerical test $\delta = 5h_{X,\Omega}$.
• Figure 5: Basis functions of Theorem \ref{['thm_pr_fast_decay_exp']}. The weight function coincides with $e^{-x^2} \in \mathcal{C}^{\infty}(\mathbb{R})$ and the approxiomation nodes are $9$ equispaced nodes in $[0,1]^2$. The basis functions reproduce the polynomials of degree $2$. In this numerical test $\delta = 5h_{X,\Omega}$.

Theorems & Definitions (13)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• Definition 4
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4