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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Radon transforms on affine Grassmannians
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by Boris Rubin PDF
Trans. Amer. Math. Soc. 356 (2004), 5045-5070 Request permission


We develop an analytic approach to the Radon transform $\hat f (\zeta )=\int _{\tau \subset \zeta } f (\tau )$, where $f(\tau )$ is a function on the affine Grassmann manifold $G(n,k)$ of $k$-dimensional planes in $\mathbb {R}^n$, and $\zeta$ is a $k’$-dimensional plane in the similar manifold $G(n,k’), \; k’>k$. For $f \in L^p (G(n,k))$, we prove that this transform is finite almost everywhere on $G(n,k’)$ if and only if $\; 1 \le p < (n-k)/(k’ -k)$, and obtain explicit inversion formulas. We establish correspondence between Radon transforms on affine Grassmann manifolds and similar transforms on standard Grassmann manifolds of linear subspaces of $\mathbb {R}^{n+1}$. It is proved that the dual Radon transform can be explicitly inverted for $k+k’ \ge n-1$, and interpreted as a direct, “quasi-orthogonal" Radon transform for another pair of affine Grassmannians. As a consequence we obtain that the Radon transform and the dual Radon transform are injective simultaneously if and only if $k+k’ = n-1$. The investigation is carried out for locally integrable and continuous functions satisfying natural weak conditions at infinity.
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Additional Information
  • Boris Rubin
  • Affiliation: Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel
  • MR Author ID: 209987
  • Email:
  • Received by editor(s): May 13, 2003
  • Received by editor(s) in revised form: September 11, 2003
  • Published electronically: June 29, 2004
  • Additional Notes: This work was supported in part by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany).

  • Dedicated: Dedicated to Professor Lawrence Zalcman on the occasion of his 60th birthday
  • © Copyright 2004 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 356 (2004), 5045-5070
  • MSC (2000): Primary 44A12; Secondary 47G10
  • DOI:
  • MathSciNet review: 2084410