Harmonic approximation by finite sums of moduli

TL;DR

This paper investigates how finite sums of harmonic functions can approximate weighted functions on the unit ball, providing solutions for weights with doubling properties and characterizations in even dimensions.

Contribution

It introduces a method to approximate weights with sums of harmonic functions and characterizes such weights in even dimensions, advancing harmonic approximation theory.

Findings

Constructed finite harmonic sums for weights with doubling property

Characterized weights allowing harmonic sum approximations in even dimensions

Provided solutions to the approximation problem for specific weight classes

Abstract

Let $h(B_d)$ denote the space of real-valued harmonic functions on the unit ball $B_d$ of $\mathbb{R}^d$, $d\ge 2$. Given a radial weight $w$ on $B_d$, consider the following problem: construct a finite family $\{f_1, f_2, \dots, f_J\}$ in $h(B_d)$ such that the sum $|f_1| + |f_2|+\dots + |f_J|$ is equivalent to $w$. We solve the problem for weights $w$ with a doubling property. Moreover, if $d$ is even, then we characterize those $w$ for which the problem has a solution.