Gaik Ambartsoumian and Peter Kuchment
On the injectivity of the circular Radon transform arising in thermoacoustic tomography
(53K, LATEX)

ABSTRACT.  The circular Radon transform integrates a function over the set of 
all spheres with a given set of centers. The problem of 
injectivity of this transform (as well as inversion formulas, 
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 M.~Agranovsky and E.~T.~Quinto. Their techniques involved 
intricate microlocal analysis and knowledge of geometry of zeros 
of harmonic polynomials in the plane, which are somewhat 
restrictive in more general circumstances. Since then there has 
been an active search for alternative methods, especially the ones 
based on simple PDE techniques. The article discusses known and 
provides new results that one can obtain by methods that 
essentially involve only the finite speed of propagation and 
domain dependence for the wave equation.