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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 $f(x) = O({\left | x \right |^{ - n - k - 1}})$ as $\left | x \right | \to \infty$. 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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Keywords: Radon transform, polynomial consistency condition, asymptotic behavior
Article copyright: © Copyright 1988 American Mathematical Society