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

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.