Nonlinear $n$-term approximation of harmonic functions from shifts of the Newtonian Kernel

TL;DR

This paper investigates the efficiency of nonlinear n-term approximation of harmonic functions within the unit ball using shifts of the Newtonian kernel, achieving optimal rates tied to harmonic Besov spaces.

Contribution

It introduces highly localized frames for Besov and Triebel-Lizorkin spaces on the sphere, enabling optimal approximation rates with shifts of the Newtonian kernel.

Findings

Established optimal approximation rates in harmonic Hardy spaces.

Constructed localized frames for Besov and Triebel-Lizorkin spaces.

Linked approximation rates to harmonic Besov space regularity.

Abstract

A basic building block in Classical Potential Theory is the fundamental solution of the Laplace equation in ${\mathbb R}^d$ (Newtonian kernel). The main goal of this article is to study the rates of nonlinear $n$-term approximation of harmonic functions on the unit ball $B^d$ from shifts of the Newtonian kernel with poles outside $\overline{B^d}$ in the harmonic Hardy spaces. Optimal rates of approximation are obtained in terms of harmonic Besov spaces. The main vehicle in establishing these results is the construction of highly localized frames for Besov and Triebel-Lizorkin spaces on the sphere whose elements are linear combinations of a fixed number of shifts of the Newtonian kernel.