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.