Range conditions for a spherical mean transform
Mark Agranovsky, David Finch, and Peter Kuchment
TL;DR
This paper characterizes the range of a spherical mean transform relevant to thermoacoustic tomography, revealing that moment conditions are unnecessary in all dimensions, simplifying the mathematical description of the transform.
Contribution
It proves that moment conditions are superfluous for the range description of the spherical mean transform in all dimensions, extending previous odd-dimensional results.
Findings
Moment conditions are unnecessary in all dimensions.
Range description includes smoothness, support, and orthogonality conditions.
Simplifies the mathematical understanding of the spherical mean transform.
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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