Completeness of asymmetric products of harmonic functions and uniqueness of the solution to the Lavrent'ev equation in inverse wave sounding problems

Abstract

We prove that the family of all pairwise products of regular harmonic functions in a domain $D \subset \mathbb{R}^3$ and Newtonian potentials of points located on a ray outside $D$ is complete in $L_2(D)$. This result is used for justification of uniqueness of a solution to the linear integral equation to which inverse problems of wave sounding in $\mathbb{R}^3$ are reduced. The corresponding inverse problems are shown to be uniquely solvable in spatially non-overdetermined settings where the dimension of the spatial data support coincides with that of the support of the sought-for function. Uniqueness theorems are used for establishing that the axial symmetry of the input data for the inverse problems under consideration implies that of the solutions to these problems.