$L^p$ Bernstein Inequalities and Inverse Theorems for RBF Approximation on $\mathbb{R}^d$

TL;DR

This paper extends Bernstein inequalities and inverse theorems for RBF approximation on Euclidean space by establishing $L^p$ bounds involving Bessel-potential norms and the separation radius.

Contribution

It introduces $L^p$ Bernstein inequalities for RBF networks on $\

Findings

Established $L^p$ Bernstein inequalities involving Bessel-potential norms.

Derived inverse theorems based on these inequalities.

Connected the bounds to the separation radius of RBF networks.

Abstract

Bernstein inequalities and inverse theorems are a recent development in the theory of radial basis function(RBF) approximation. The purpose of this paper is to extend what is known by deriving $L^p$ Bernstein inequalities for RBF networks on $\mathbb{R}^d$. These inequalities involve bounding a Bessel-potential norm of an RBF network by its corresponding $L^p$ norm in terms of the separation radius associated with the network. The Bernstein inequalities will then be used to prove the corresponding inverse theorems.