On the Completeness of Products of Harmonic Functions and the Uniqueness of the Solution of the Inverse Acoustic Sounding Problem

Abstract

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 ⊂ Rⁿ is complete in L₂(D), where D is a bounded domain in Rⁿ, n ≥ 3, such that \(\bar D\) ∩ L = ∅. This result is used in the proof of uniqueness theorems for the inverse acoustic sounding problem in R³.