  In this paper, we provide a simple framework to derive and analyse several
classes of effective one-step methods. The framework consists in the
discretization of a local Fourier expansion of the continuous problem.
Different choices of the basis lead to different classes of methods, even
though we shall here consider only the case of an orthonormal polynomial basis,
from which a large subclass of Runge-Kutta methods is derived. The obtained
results are then applied to prove, in a simplified way, the order and stability
properties of Hamiltonian BVMs (HBVMs), a recently introduced class of energy
preserving methods for canonical Hamiltonian systems. A few numerical tests
with such methods are also included, in order to confirm the effectiveness of
the methods.
