On the Fourier coefficients of the perturbation written using the angle variable near a separatrix loop

For a Hamiltonian system with one degree of freedom with the Hamiltonian depending on a parameter we consider an arbitrary analytic function that depends on the phase point, the parameter, and also is periodically dependent on the time. We prove an estimate for the Fourier coefficients of this function as a periodic function of the angle (from the pair of action-angle variables) and time. This estimate is important to study time-periodic perturbations of such systems, as such functions arise naturally as the coefficients of the perturbed system written using the angle variable. Having an estimate for the Fourier coefficients is needed for the application of the averaging method and for the study of the resonances.

The estimate we prove is most likely known to experts. However, we were unable to find a reference for it and it is crucial to the research project ''Adiabatic invariance in two-frequency dynamical systems with separatrix crossing'' we are working on. This is why we have written down its proof as this preprint.