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.
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUS-CI-016-04
Acknowledgements: The work of the first author was partially supported by the ISF (Israel Science Foundation) Grant 688/08 and by the Texas A&M University. The third author was partially supported by the NSF grant DMS 0604778 and by the KAUST grant KUS-CI-016-04. The authors express their gratitude to NSF, Texas A&M University, and KAUST for the support. The authors also thank Y. Lyubarskii and L. Nguyen for discussions and the referee for useful remarks.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.