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Transactions of the American Mathematical Society

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



The dependence of the generalized Radon transform on defining measures

Author: Eric Todd Quinto
Journal: Trans. Amer. Math. Soc. 257 (1980), 331-346
MSC: Primary 58G15; Secondary 44A99
MathSciNet review: 552261
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Abstract: Guillemin proved that the generalized Radon transform R and its dual ${R^t}$ are Fourier integral operators and that ${R^t}R$ is an elliptic pseudodifferential operator. In this paper we investigate the dependence of the Radon transform on the defining measures. In the general case we calculate the symbol of ${R^t}R$ as a pseudodifferential operator in terms of the measures and give a necessary condition on the defining measures for ${R^t}R$ to be invertible by a differential operator. Then we examine the Radon transform on points and hyperplanes in ${\textbf {R}^n}$ with general measures and we calculate the symbol of ${R^t}R$ in terms of the defining measures. Finally, if ${R^t}R$ is a translation invariant operator on ${\textbf {R}^n}$ then we prove that ${R^t}R$ is invertible and that our condition is equivalent to ${({R^t}R)^{ - 1}}$ being a differential operator.

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Keywords: Generalized Radon transforms, Radon transforms on <!– MATH ${\textbf {R}^n}$ –> <IMG WIDTH="34" HEIGHT="22" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${\textbf {R}^n}$">, smooth positive measure, pseudodifferential operator, symbol of pseudodifferential operator
Article copyright: © Copyright 1980 American Mathematical Society