  We define a Mellin amplitude for CFT$_1$ four-point functions. Its analytical
properties are inferred from physical requirements on the correlator. We
discuss the analytic continuation that is necessary for a fully nonperturbative
definition of the Mellin transform. The resulting bounded, meromorphic function
of a single complex variable is used to derive an infinite set of
nonperturbative sum rules for CFT data of exchanged operators, which we test on
known examples. We then consider the perturbative setup produced by quartic
interactions with an arbitrary number of derivatives in a bulk AdS$_2$ field
theory. With our formalism, we obtain a closed-form expression for the Mellin
transform of tree-level contact interactions and for the first correction to
the scaling dimension of "two-particle" operators exchanged in the generalized
free field theory correlator.
