Geodesic
On the Completeness of Products of Harmonic Functions and the Uniqueness of the Solution of the Inverse Acoustic Sounding Problem
Matematičeskie zametki, Tome 104 (2018) no. 5, pp. 708-716

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It is proved that the family of all pairwise products of regular harmonic functions on $D$ and of the Newtonian potentials of points on the line $L\subset\mathbb R^n$ is complete in $L_2(D)$, where $D$ is a bounded domain in $\mathbb R^n$, $n\ge 3$, such that $\overline D\cap L=\varnothing$. This result is used in the proof of uniqueness theorems for the inverse acoustic sounding problem in $\mathbb R^3$.
Keywords: harmonic function, Newtonian potential, completeness, integral equation, acoustic sounding, inverse problem, unique solvability. 
M. Yu. Kokurin. On the Completeness of Products of Harmonic Functions and the Uniqueness of the Solution of the Inverse Acoustic Sounding Problem. Matematičeskie zametki, Tome 104 (2018) no. 5, pp. 708-716. http://geodesic.mathdoc.fr/item/MZM_2018_104_5_a7/
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