On expansions over harmonic polynomial products in ${\mathbb R}^3$

In inverse problems, an important role is played by the following fact: the functions of the form
\begin{align*} \sum_{k=1}^{n} f_k(x,y,z) g_k(x,y,z), \end{align*}
where $f_k,g_k$ are the solutions of a second order elliptic equation in a bounded domain $\Omega\subset\mathbb R^3$, constitute a dense set in $L_2(\Omega)$.
This paper deals with the Laplace equation. We show that the density does hold if $f_k$ and $g_k$ are harmonic polynomials, whereas the factors $g_k$ are invariant with respect to shifts or rotations.