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For the defocusing Nonlinear Schr\"odinger equation on the circle, weconstruct a Birkhoff map $\Phi$ which is tame majorant analytic in aneighborhood of the origin. Roughly speaking, majorant analytic means thatreplacing the coefficients of the Taylor expansion of $\Phi$ by their absolutevalues gives rise to a series (the majorant map) which is uniformly andabsolutely convergent, at least in a small neighborhood. Tame majorant analyticmeans that the majorant map of $\Phi$ fulfills tame estimates. The proof isbased on a new tame version of the Kuksin-Perelman theorem, which is aninfinite dimensional Vey type theorem.

Tame majorant analyticity for the Birkhoff map of the defocusing Nonlinear Schr\"odinger equation on the circle