# On sound ranging in Hilbert space

## TL;DR

This paper investigates the sound ranging problem in infinite-dimensional Hilbert space, providing conditions for solution uniqueness and examples illustrating multiple solutions, extending classical methods to an abstract mathematical setting.

## Contribution

It introduces solving methods for sound ranging in Hilbert spaces and establishes conditions for solution uniqueness and multiplicity, which are novel in this infinite-dimensional context.

## Key findings

- Provided sufficient conditions for uniqueness of solutions.
- Constructed examples with basis sensor sets illustrating solution multiplicity.
- Extended sound ranging problem analysis to infinite-dimensional spaces.

## Abstract

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