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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the range of the Radon $ d$-plane transform and its dual


Author: Fulton B. Gonzalez
Journal: Trans. Amer. Math. Soc. 327 (1991), 601-619
MSC: Primary 44A12; Secondary 43A85, 92C55
MathSciNet review: 1025754
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Abstract: We present direct, group-theoretic proofs of the range theorem for the Radon $ d$-plane transform $ f \to \hat f$ on $ \mathcal{S}({\mathbb{R}^n})$. (The original proof, by Richter, involves extensive use of local coordinate calculations on $ G(d,n)$, the Grassmann manifold of affine $ d$-planes in $ {\mathbb{R}^n}$.) We show that moment conditions are not sufficient to describe this range when $ d < n - 1$, in contrast to the compactly supported case. Finally, we show that the dual $ d$-plane transform maps $ \mathcal{E}(G(d,n))$ surjectively onto $ \mathcal{E}({\mathbb{R}^n})$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1991-1025754-4
PII: S 0002-9947(1991)1025754-4
Keywords: Radon $ d$-plane transform, range theorem, moment conditions, Euclidean motion group, dual Radon transform, left regular representation
Article copyright: © Copyright 1991 American Mathematical Society



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