Radial polynomials as alternatives to smooth radial basis functions and their applications

TL;DR

This paper introduces radial polynomials as an alternative to smooth radial basis functions, aiming to match their approximation power while simplifying shape parameter selection.

Contribution

It proposes radial polynomials that overcome shape parameter tuning issues and achieve comparable approximation accuracy to traditional smooth RBFs.

Findings

Radial polynomials demonstrate similar approximation accuracy to smooth RBFs.

The proposed method simplifies the shape parameter selection process.

Radial polynomials enhance computational efficiency in function approximation.

Abstract

Because of the high approximation power and simplicity of computation of smooth radial basis functions (RBFs), in recent decades they have received much attention for function approximation. These RBFs contain a shape parameter that regulates the relation between their accuracy and stability. A difficulty in approximation via smooth RBFs is optimal selection of shape parameter. The aim of this paper is to introduce an alternative for smooth RBFs, which in addition to overcoming this difficulty, its approximation power is almost equal to RBFs....