  A slightly modified variant of the cubic periodic one-dimensional nonlinear
Schroedinger equation is shown to admit weak solutions for all initial data in
certain function spaces wider than L^2. These solutions depend uniformly
continuously on the initial data, in the norms considered. The solutions are
constructed as sums of infinite series of multilinear operators applied to
initial data; no fixed point argument or energy inequality are used. In a
companion paper we have shown that weak solutions in these same function spaces
are however not unique.
