  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.
