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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
DOI: https://doi.org/10.1090/S0002-9947-1980-0552261-8
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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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1980-0552261-8
Keywords: Generalized Radon transforms, Radon transforms on $ {\textbf{R}^n}$, smooth positive measure, pseudodifferential operator, symbol of pseudodifferential operator
Article copyright: © Copyright 1980 American Mathematical Society

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