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Dual Radon transforms on affine Grassmann manifolds

Authors: Fulton B. Gonzalez and Tomoyuki Kakehi
Journal: Trans. Amer. Math. Soc. 356 (2004), 4161-4180
MSC (2000): Primary 44A12; Secondary 43A85
Published electronically: April 16, 2004
MathSciNet review: 2058842
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Abstract: Fix $0 \leq p < q \leq n-1$, and let $G(p,n)$ and $G(q,n)$denote the affine Grassmann manifolds of $p$- and $q$-planes in $\mathbb{R} ^n$. We investigate the Radon transform $\mathcal{R}^{(q,p)} : C^{\infty} (G(q,n)) \to C^{\infty} (G(p,n))$associated with the inclusion incidence relation. For the generic case $\dim G(q,n) < \dim G(p,n)$ and $ p+q > n$, we will show that the range of this transform is given by smooth functions on $G(p,n)$ annihilated by a system of Pfaffian type differential operators. We also study aspects of the exceptional case $p+q =n$.

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Additional Information

Fulton B. Gonzalez
Affiliation: Department of Mathematics, Tufts University, Medford, Massachusetts 02155-7049

Tomoyuki Kakehi
Affiliation: Institute of Mathematics, University of Tsukuba, Ibaraki, Japan 305-8571

Keywords: Radon transform, Grassmannian, Pfaffian systems
Received by editor(s): November 26, 2002
Received by editor(s) in revised form: May 1, 2003, and July 17, 2003
Published electronically: April 16, 2004
Article copyright: © Copyright 2004 American Mathematical Society