Harmonic Polynomials Via Differentiation

Abstract

It is well-known that if p is a homogeneous polynomial of degree k in n variables, p ∈ P k , then the ordinary derivative p(∇) r 2−n has the form A n,k Y(x)r 2−n−2k where A n,k is a constant and where Y is a harmonic homogeneous polynomial of degree k, Y ∈ H k , actually the projection of p onto H k. Here we study the distributional derivative p ∇ r 2−n and show that the ordinary part is still a multiple of Y, but that the delta part is independent of Y, that is, it depends only on p−Y. We also show that the exponent 2−n is special in the sense that the corresponding results for p(∇)(r α) do not hold if α = 2−n. Furthermore, we establish that harmonic polynomials appear as multiples of r 2−n−2k−2k when p(∇) is applied to harmonic multipoles of the form Y (x)r 2−n−2k for some Y ∈ H k .