The main problem addressed in the talk is the characterization of Fantappie transform of positive measures in the unit ball of C\\^n. The analogous real results were obtained by G.M.Henkin and A.A. Shananin, in the line of the classical theorem of Bernstein on the line. I will propose an approach based on Hilbert spaces of analytic functions. This will provide, among other things, a novel proof of Martineau\'s duality theorem. Based on joint work in progress with John McCarthy.