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

2018/08/26 by Kamen G. Ivanov, Ivanov, Kamen, Pencho Petrushev +1
Mathematics · #Advanced Harmonic Analysis Research

paper · pdf · doi:10.48550/arxiv.1808.08641

Abstract

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

Related

• Nonlinear approximation of harmonic functions from shifts of the Newtonian kernel in BMO
• Highly localized kernels on the sphere induced by Newtonian kernels
• M-harmonic reproducing kernels on the ball
• H-harmonic reproducing kernels on the ball
• Sub-exponentially localized kernels and frames induced by orthogonal expansions