  For the defocusing Nonlinear Schr\"odinger equation on the circle, we
construct a Birkhoff map $\Phi$ which is tame majorant analytic in a
neighborhood of the origin. Roughly speaking, majorant analytic means that
replacing the coefficients of the Taylor expansion of $\Phi$ by their absolute
values gives rise to a series (the majorant map) which is uniformly and
absolutely convergent, at least in a small neighborhood. Tame majorant analytic
means that the majorant map of $\Phi$ fulfills tame estimates. The proof is
based on a new tame version of the Kuksin-Perelman theorem, which is an
infinite dimensional Vey type theorem.
