On sound ranging in Hilbert space

Abstract

We consider the sound ranging problem (SRP), which is to find the position of source-point from the moments when spherical wave reaches sensor-points, in the infinite-dimensional separable Hilbert space, and describe the solving methods, for entire space and for its unit sphere. In the former case, we give some sufficient conditions for solution's uniqueness. We also provide two examples with the sets of sensors being a basis: 1st, when SRP and so-called dual problem both have single solutions, and 2nd, when SRP has two distinct solutions.