The linear Vlasov-Poisson system for homogeneous, stable equilibria is solved by means of a novel integral transform that is a generalization of the Hilbert transform. The integral transform provides a means for describing the dynamics of the continuous spectrum that is well-known to occur in this system. The results are interpreted in the context of Hamiltonian systems theory, where it is shown that the integral transform defines a canonical transformation to action-angle variables for this infinite degree-of-freedom system. A means for attaching Kren (energy) signature to a continuum eigenmode is given.