Harmonic Polynomials via Differentiation (Analysis in Theory and Applications, 2018)
It is well-known that if $p$ is a homogeneous polynomial of degree $k$ in $n$ variables, $p∈\mathcal{P}_k$, then the ordinary derivative $p(∇) (r^{2−n})$ has the form $A_{n,k}Υ(x)r^{2−n−2k}$ where $A_{n,k}$ is a constant and where $Υ$ is a harmonic homogeneous polynomial of degree $k,$ $Υ\in \mathcal{H}_k$, actually the projection of $p$ onto $\mathcal{H}_k$. Here we study the distributionalderivative $p(\overline{∇})(r^{2−n})$ and show that the ordinary part is still a multiple of $Υ$, but thatthe delta part is independent of $Υ$, that is, it depends only on $p−Υ$. We also show thatthe exponent $2−n$ is special in the sense that the corresponding results for $p(∇)(r^α)$ do not hold if $α eq 2−n$.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 $Υ'(x)r^{2−n−2k'}$ for some $Υ' \in \mathcal{H}_k$.
Keywords: Harmonic functions; harmonic polynomials; distributions; multipoles.