Some Personal Remarks on the Radon Transform

Back in 1955, I worked on extending the mean value theorem of Leifur Ásgeirsson [1937] for the ultrahyperbolic equation on Rn ×Rn to Riemannian homogeneous spaces G/K ×G/K. I was motivated by Godement’s generalization [1952] of the mean value theorem for Laplace’s equation Lu = 0 to the system Du = 0 for all G-invariant differential operators D (annihilating the constants) on G/K. At the time (Spring 1955) I visited Leifur in New Rochelle where he was living in the house of Fritz John (then on leave from NYU). They had both been students of Courant in Göttingen in the 1930’s. Since John’s book [1955] treats Ásgeirsson’s theorem in some detail, Leifur lent me a copy of it (in page proofs) to look through on the train to Boston. I was quickly enticed by Radon’s formulas (in John’s formulation) for a function f on Rn in terms of its integrals over hyperplanes. In John’s notation, let J(ω, p) denote the integral of f over the hyperplane