Direct method for a finitely smooth problem with ``small denominators'' (Q923808)

A direct method is proposed for obtaining estimates for the formal Fourier series expansions which occur in the investigation of linear systems with quasi-periodic coefficients in the finitely smooth (i.e. nonanalytic) case. The method applies to systems of the type \(\dot X=\lambda P(t)X\) where \[ P(t)=\sum_{n>0}\sum_{m\geq 0}P_{nm} \exp (it((n,\omega)-m)), \] n\(=(n_ 1,...,n_{\ell})\), \(P_{nm}\) are constant quadratic matrices with \(\| P_{nm}\| \leq (\| n\| +| m|)^{-\gamma},\gamma >1\); \(\omega =(\omega_ 1,...,\omega_{\ell})\) is a vector with real components \(\omega_ i>0\) which are linearly independent on the field of real numbers; \(\lambda\) is a complex parameter which has to be sufficiently small.