Let $\mathcal{H}$ denote the space of all harmonic polynomials with complex coefficients in $n$ variables $x_1,\ldots, x_n$ in $\mathbb{R}^n$. I'm trying to show that the linear span of the set $\mathcal{M}=\{p\, q: p,q\in\mathcal{H}\}$ is dense in $C(K)$ under supremum norm, where $K$ is a compact set in $\mathbb{R}^n$, for $n\geq 2$. One may assume that $K$ is the closed unit ball in $\mathbb{R}^n$ since once it is proved for the unit ball, the result follows for any compact $K$. This seems to be classical and should be known but I have not found a reference.

When $n$ is even, it is not hard to prove. In fact, if $n=2m$, one may identify $\mathbb{R}^n$ with $\mathbb{C}^{m}$ and use the fact that the linear span of $\mathcal{N}=\{f\,\overline{g}: f, g \text{ are holomorphic polynomials}\}$ is dense in $C(K)$ by Stone-Weierstrass Theorem. Note that for any holomorhpic polynomials $f, g$ in $\mathbb{C}^m$, the polynomials $p=f$ and $q= \overline{g}$ are harmonic in $\mathbb{R}^n$. Therefore, $\mathcal{N}\subset\mathcal{M}$ and hence, $\mathcal{M}$ is dense in $C(K)$.

I have not been able to resolve the case $n$ is odd. In general, $\mathcal{M}$ is **not** a subalgebra of $C(K)$ even though it is self-adjoint, separates points and vanishes at no point so Stone-Weierstrass Theorem cannot apply directly to $\mathcal{M}$ (even when $n$ is even).

I would highly appreciate it if someone can point out a reference or offer a proof.