On the injectivity of the circular Radon transform

The circular Radon transform integrates a function over the set of all spheres with a given set of centres. The problem of injectivity of this transform (as well as inversion formulae, range descriptions, etc) arises in many fields from approximation theory to integral geometry, to inverse problems for PDEs and recently to newly developing types of tomography. A major breakthrough in the 2D case was made several years ago in a work by Agranovsky and Quinto. Their techniques involved microlocal analysis and known geometric properties of zeros of harmonic polynomials in the plane. Since then there has been an active search for alternative methods, especially those based on simple PDE techniques, which would be less restrictive in more general situations. This paper provides some new results that one can obtain by methods that essentially involve only the finite speed of propagation and domain dependence for the wave equation.