Inversion of radon transforms using wavelet transforms generated by wavelet measures

The classical inversion formulae for the Radon transform R on Rⁿ involve the dual transform R^# and the operator D^n-1 = (-Δ)^(n-1)/2 which represents the positive power of the Laplacian. It is shown that wavelet type representations of D^n-1, generated by finite measures with a certain number of vanishing moments, lead to explicit inversion formulae for Rf, f ∈ L^p(Rⁿ), and enable one to characterize the range R(L^p(Rⁿ)). The same method can be applied to explicit inversion and characterization of Radon transforms of finite Borel measures.