Unitarization of the Radon transform on homogeneous trees
During 20th century, the problem of inverting Radon transform has been deeply studied because of its various applications. A classical use in the continuous case (e.g. two-dimensional signals) is the medical tomography. Recently, Radon on discrete setups (e.g. graphs) also began to be treated and network tomography has been developed with the same philosophy of algorithms used in CT scan. We analyze Radon on a discrete manifold: a homogeneous tree X. On X the set H of all the horocycles is defined, they are the analog of hyperplanes in Euclidean spaces. Given a signal f on X, its Radon transform Rf is defined at a horocycle h as the sum of values of f at vertices lying on h. We solves the unitarization problem: we show the existence of a pseudo-differential operator such that its postcomposition with R extends to a unitary operator Q from L^2(X) to L^2(H). Furthermore, such Q is an intertwining operator for the representations of the automorphism group of X on L^2(X) and the one on L^2(H), which are therefore unitary equivalent.
2020-04-28 at 3:00 pm