The dependence of the generalized Radon transform on defining measures

Quinto, Eric Todd

doi:10.1090/S0002-9947-1980-0552261-8

The dependence of the generalized Radon transform on defining measures
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by Eric Todd Quinto

Trans. Amer. Math. Soc. 257 (1980), 331-346

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

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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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