Completeness of products of homogeneous harmonic polynomials and uniqueness of the solution to an inverse wave sounding problem

We prove that pairwise products of elements from two classes of homogeneous harmonic polynomials (HHPs) are complete in L2(D), where D is a bounded domain in R3. One of the classes is formed by all HHPs, the second class contains one HHP of each degree l=0,1,…. The result reinforces a long–standing theorem by A. Calderon (1980), in which both classes consisted of all HHPs. The strengthened Calderon's theorem is used to substantiate uniqueness of the solution to an inverse wave sounding problem in the spatially nonoverdetermined setting.