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

Gonzalez, Fulton B.

doi:10.1090/S0002-9947-1991-1025754-4

On the range of the Radon $d$-plane transform and its dual
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Trans. Amer. Math. Soc. 327 (1991), 601-619 Request permission

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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