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. 
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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/

[1] A.P. Calderon, “On an inverse boundary value problem”, Seminar on Numerical Analysis and Its Applications to Continuum Physics, 1980, 65–73

[2] M. Yu. Kokurin, “On the Completeness of Products of Harmonic Functions and the Uniqueness of the Solution of the Inverse Acoustic Sounding Problem”, Math. Notes, 104:5 (2018), 689–695 | DOI | MR | Zbl

[3] M. I. Belishev, A. F. Vakulenko, “Ob algebre garmonicheskikh kvaternionnykh polei v ${\mathbb R}^3$”, Algebra i analiz, 31:1 (2019), 1–17

[4] V. Isakov, Inverse Problems for Partial Differential Equations, Springer, N.Y., 2006 | Zbl

[5] A. N. Tikhonov, A. A. Samarskii, Uravneniya matematicheskoi fiziki, Nauka, M., 1966