The dependence of the generalized Radon transform on defining measures

Abstract

Guillemin proved that the generalized Radon transform R and its dual R t {R^t} are Fourier integral operators and that R t R {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 {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 {R^t}R to be invertible by a differential operator. Then we examine the Radon transform on points and hyperplanes in R n {\textbf {R}^n} with general measures and we calculate the symbol of R t R {R^t}R in terms of the defining measures. Finally, if R t R {R^t}R is a translation invariant operator on R n {\textbf {R}^n} then we prove that R t R {R^t}R is invertible and that our condition is equivalent to ( R t R ) − 1 {({R^t}R)^{ - 1}} being a differential operator.