The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds

As proved by Radon [16] and John [13], a differentiable function / of compact support on a Euclidean space R n can be determined explicitly by means of its integrals over the hyperplanes in the space.Let J(~o, p) denote the integral of / over the hyperplane <x, co> =p where r is a unit vector and <,> the inner product in R n.If A denotes the Laplacian on R n, do) the area element on the unit sphere S n-1 then (John [14], p. 13) J(x) =1(2~i)i-n (Ax)~(n-1)fs.J(~, <eo, x))dw, (n odd);(1)where, in the last formula, the Cauchy principal value is taken.Considering now the simpler formula (1) we observe that it contains two dual integrations: the first over the set of points in a given hyperplane, the second over the set of hyperplanes passing through a given point.Generalizing this situation we consider the following setup:(i) Let X be a manifold and G a transitive Lie transformation group of X.Let ~ be a family of subsets of X permuted transitively by the action of G on X, whence ~ acquires a G-invariant differentiable structure.Here ~ will be called the dual space of X.(ii) Given x E X, let ~ denote the set of ~ e 7~ passing through x.It is assumed that each and each ~ carry measures ju and v, respectively, such that the action of G on X and permutes the measures # and permutes the measures v.