Reconstruction of functions from their integrals over k-planes

The k-plane Radon transform assigns to a function f(x) on ℝⁿ the collection of integrals f̂(τ) = ∫_τ f over all k-dimensional planes τ. We give a systematic treatment of two inversion methods for this transform, namely, the method of Riesz potentials, and the method of spherical means. We develop new analytic tools which allow to invert f̂(τ) under minimal assumptions for f. It is assumed that f ∈ L^p, 1 ≤ p < n/k, or f is a continuous function with minimal rate of decay at infinity. In the framework of the first method, our approach employs intertwining fractional integrals associated to the k-plane transform. Following the second method, we extend the original formula of Radon for continuous functions on ℝ²2 to f ∈ L^p(ℝⁿ) and all 1 ≤ k < n. New integral formulae and estimates, generalizing those of Fuglede and Solmon, are obtained.