04-129 Gaik Ambartsoumian and Peter Kuchment

On the injectivity of the circular Radon transform arising in thermoacoustic tomography (53K, LATEX) Apr 26, 04

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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.

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