The spectral properties of distributions and asymptotic methods in perturbation theory

For differential equations of the form x' = epsilon f(t, x; epsilon) in a Banach space a modification of the classical Krylov-Bogolyubov method is put forward. It allows complications in the construction of higher-order approximations which stem from the 'small denominators problem' to be avoided and many of the standard constraints on the behaviour of the function f to be eliminated. The approach suggested is based on some results on the Fourier transforms of distributions.