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

 

X-rays of forms and projections of currents


Author: Bruce Solomon
Journal: Trans. Amer. Math. Soc. 363 (2011), 143-164
MSC (2010): Primary 44A12, 42A85, 58A10, 58A25
Published electronically: August 31, 2010
MathSciNet review: 2719676
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Abstract: We study a new Radon-like transform that averages projected $ p$-forms in $ \mathbf R^{n}$ over affine $ (n-k)$-spaces. We then prove an explicit inversion formula for our transform on the space of rapidly-decaying smooth $ p$-forms. Our transform differs from the one in the work by Gelfand, Graev and Shapiro (1969). Moreover, if it can be extended to a somewhat larger space of $ p$-forms, our inversion formula will allow the synthesis of any rapidly-decaying smooth $ p$-form on $ \mathbf R^{n} $ as a (continuous) superposition of pullbacks from $ p$-forms on $ k$-dimensional subspaces. In turn, such a synthesis implies an explicit formula (which we derive) for reconstructing compactly supported currents in $ \mathbf R^{n}$ (e.g., compact oriented $ k$-dimensional subvarieties) from their oriented projections onto $ k$-planes.


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Additional Information

Bruce Solomon
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Email: solomon@indiana.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05348-6
PII: S 0002-9947(2010)05348-6
Received by editor(s): July 22, 2008
Published electronically: August 31, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.