Geodesic
On expansions over harmonic polynomial products in~${\mathbb R}^3$
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 51, Tome 506 (2021), pp. 36-42

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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. 
@article{ZNSL_2021_506_a4,
     author = {A. F. Vakulenko},
     title = {On expansions over harmonic polynomial products in~${\mathbb R}^3$},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {36--42},
     publisher = {mathdoc},
     volume = {506},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2021_506_a4/}
}
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A. F. Vakulenko. On expansions over harmonic polynomial products in~${\mathbb R}^3$. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 51, Tome 506 (2021), pp. 36-42. http://geodesic.mathdoc.fr/item/ZNSL_2021_506_a4/