August  2009, 3(3): 373-382. doi: 10.3934/ipi.2009.3.373

Range conditions for a spherical mean transform

1. 

Mathematics Department, Bar Ilan University, Ramat Gan 52900, Israel

2. 

Mathematics Department, Oregon State University, Corvallis, OR 97331-4605, United States

3. 

Mathematics Department, Texas A&M University, College Station, TX 77843-3368, United States

Received  February 2009 Revised  May 2009 Published  July 2009

The paper is devoted to the range description of the Radon type transform that averages a function over all spheres centered on a given sphere. Such transforms arise naturally in thermoacoustic tomography, a novel method of medical imaging. Range descriptions have recently been obtained for such transforms, and consisted of smoothness and support conditions, moment conditions, and some additional orthogonality conditions of spectral nature. It has been noticed that in odd dimensions, surprisingly, the moment conditions are superfluous and can be eliminated. It is shown in this text that in fact the same happens in any dimension.
Citation: Mark Agranovsky, David Finch, Peter Kuchment. Range conditions for a spherical mean transform. Inverse Problems & Imaging, 2009, 3 (3) : 373-382. doi: 10.3934/ipi.2009.3.373
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