Polyharmonic approximation on the sphere

TL;DR

This paper improves error estimates for spherical approximation using polyharmonic Green's functions, showing optimal approximation order for functions with certain smoothness levels, enhancing previous theoretical bounds.

Contribution

It provides new, sharper error estimates for SBF approximation on the sphere, extending the approximation order results to a broader range of $L_p$ spaces.

Findings

Achieves $L_p$ approximation order $\sigma$ for functions with $L_p$ smoothness $\sigma$

Extends error estimates to $p>2$ and removes previous restrictions for $p<2$

Improves theoretical understanding of spherical polyharmonic approximation

Abstract

The purpose of this article is to provide new error estimates for a popular type of SBF approximation on the sphere: approximating by linear combinations of Green's functions of polyharmonic differential operators. We show that the $L_p$ approximation order for this kind of approximation is $\sigma$ for functions having $L_p$ smoothness $\sigma$ (for $\sigma$ up to the order of the underlying differential operator, just as in univariate spline theory). This is an improvement over previous error estimates, which penalized the approximation order when measuring error in $L_p$, p>2 and held only in a restrictive setting when measuring error in $L_p$, p<2.