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

 

Radon's inversion formulas


Author: W. R. Madych
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 4475-4491
MSC (2000): Primary 44A12, 42B25
Published electronically: January 16, 2004
MathSciNet review: 2067130
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Abstract: Radon showed the pointwise validity of his celebrated inversion formulas for the Radon transform of a function $f$ of two real variables (formulas (III) and (III$'$) in J. Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Ber. Verh. Sächs. Akad. Wiss. Leipzig, Math.-Nat. kl. 69 (1917), 262-277) under the assumption that $f$ is continuous and satisfies two other technical conditions. In this work, using the methods of modern analysis, we show that these technical conditions can be relaxed. For example, the assumption that $f$ be in $L^p(\mathbb{R} ^2)$for some $p$ satisfying $4/3<p<2$ suffices to guarantee the almost everywhere existence of its Radon transform and the almost everywhere validity of Radon's inversion formulas.


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Additional Information

W. R. Madych
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009
Email: madych@uconn.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03404-X
PII: S 0002-9947(04)03404-X
Received by editor(s): May 12, 2003
Published electronically: January 16, 2004
Article copyright: © Copyright 2004 American Mathematical Society