Range conditions for a spherical mean transform

Mark Agranovsky; David Finch; Peter Kuchment

doi:10.3934/ipi.2009.3.373

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2009, Volume 3, Issue 3: 373-382. Doi: 10.3934/ipi.2009.3.373

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Range conditions for a spherical mean transform

1.
Mathematics Department, Bar Ilan University, Ramat Gan 52900
2.
Mathematics Department, Oregon State University, Corvallis, OR 97331-4605
3.
Mathematics Department, Texas A&M University, College Station, TX 77843-3368

Received: January 31, 2009

Revised: April 30, 2009

Published: July 2009

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Abstract

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.

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Mathematics Subject Classification: Primary: 44A12, 35Q99; Secondary: 92C55.

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