The theory of modulated Fourier expansions is a powerful tool for the study of the long-time behaviour of differential equations with highly oscillatory solutions (conservation of energy, momentum, and harmonic actions).
There is a discrete analogue that permits to study the long-time behaviour of linear multistep methods applied to (non-oscillatory) Hamiltonian systems. The parasitic solutions of the difference equations play the role of harmonic oscillations.
In this talk we explain the common ideas of both theories. This is joint-work with Christian Lubich.