  Recently, the numerical solution of stiffly/highly-oscillatory Hamiltonian
problems has been attacked by using Hamiltonian Boundary Value Methods (HBVMs)
as spectral methods in time. While a theoretical analysis of this spectral
approach has been only partially addressed, there is enough numerical evidence
that it turns out to be very effective even when applied to a wider range of
problems. Here we fill this gap by providing a thorough convergence analysis of
the methods and confirm the theoretical results with the aid of a few numerical
tests.
