Using the gravity side of the AdS/CFT correspondence, we investigate the analytic properties of thermal retarded Green's functions for scalars, conserved currents, the stress tensor, and massless fermions. We provide some results concerning their large frequency behavior (introducing a novel contraction mapping), their small frequency behavior and their pole structure. From these results, it is straightforward to prove the validity of various sum rules on the field theory side of the duality. As a by-product we prove some statements about existence and restrictions on quasi-normal modes in asymptotically anti-deSitter spacetimes.
