  An approximation Ansatz for the operator solution, $U(z',z)$, of a hyperbolic
first-order pseudodifferential equation, $\d_z + a(z,x,D_x)$ with $\Re (a) \geq
0$, is constructed as the composition of global Fourier integral operators with
complex phases. An estimate of the operator norm in $L(H^{(s)},H^{(s)})$ of
these operators is provided which allows to prove a convergence result for the
Ansatz to $U(z',z)$ in some Sobolev space as the number of operators in the
composition goes to $\infty$.
