The Dependence of the Generalized Radon Transform on Defining Measures

Guillemin proved that the generalized Radon transform R and its dual Rᵗ are Fourier integral operators and that RᵗR 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 RᵗR as a pseudodifferential operator in terms of the measures and give a necessary condition on the defining measures for RᵗR to be invertible by a differential operator. Then we examine the Radon transform on points and hyperplanes in Rⁿ with general measures and we calculate the symbol of RᵗR in terms of the defining measures. Finally, if RᵗR is a translation invariant operator on Rⁿ then we prove that RᵗR is invertible and that our condition is equivalent to (RᵗR)- 1 being a differential operator.