“…First notice that since X ∈ X ν with ν > 1, the Fourier coefficients are decaying exponentially fast to 0, and thus the functions v, w, p and s are well defined and smooth (in fact analytic) 2-periodic functions. Then, having F (X) = 0 means exactly that the sequences (v k ) k≥0 , (w k ) k≥0 , (p k ) k≥0 and (s k ) k≥0 solve (15), which in turn implies that the functions v, w, p and s solve (13). All the derivatives needed in (13) are legitimate thanks to the exponential decay of the coefficients.…”