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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A range theorem for the Radon transform


Authors: W. R. Madych and D. C. Solmon
Journal: Proc. Amer. Math. Soc. 104 (1988), 79-85
MSC: Primary 44A15; Secondary 26B40
DOI: https://doi.org/10.1090/S0002-9939-1988-0958047-7
MathSciNet review: 958047
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Abstract: Conditions are prescribed for a function $ g$ which are sufficient to ensure that it is the Radon transform of a continuous function $ f$ on $ {{\mathbf{R}}^n}$ such that $ f(x) = O({\left\vert x \right\vert^{ - n - k - 1}})$ as $ \left\vert x \right\vert \to \infty $. Roughly speaking, these criteria involve smoothness and the classical polynomial consistency conditions up to order $ k$ on $ g$. In particular, the result implies Helgason's Schwartz theorem for the Radon transform [Acta Math. 113 (1965)].


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DOI: https://doi.org/10.1090/S0002-9939-1988-0958047-7
Keywords: Radon transform, polynomial consistency condition, asymptotic behavior
Article copyright: © Copyright 1988 American Mathematical Society