Nonlinear approximation of harmonic functions from shifts of the Newtonian kernel in BMO

TL;DR

This paper develops a method for nonlinear approximation of harmonic functions using shifts of the Newtonian kernel, establishing sharp estimates in Besov spaces through localized frames.

Contribution

It introduces a novel approach to nonlinear approximation in harmonic analysis using localized frames of shifts of the Newtonian kernel in BMO and Besov spaces.

Findings

Established a sharp Jackson estimate involving Besov spaces.

Constructed localized frames for Besov and VMO spaces on the sphere.

Demonstrated effective approximation of harmonic functions using kernel shifts.

Abstract

We study nonlinear n-term approximation of harmonic functions on the unit ball in $R^d$ from linear combinations of shifts of the Newtonian kernel (fundamental solution of the Laplace equation) in BMO. A sharp Jackson estimate is established that naturally involves certain Besov spaces. The method for obtaining this result is based on the construction of highly localized frames for Besov spaces and VMO on the sphere whose elements are linear combinations of a fixed number of shifts of the Newtonian kernel.