The authors introduce classes of non-quasianalytic functions \({\mathcal E}_{\omega}({\mathbb{R}}^ n)\) similar to those treated by Beurling and Björck: Given a weight function \(\omega\) : \({\mathbb{R}}\to [0,\infty [\) (i.e. \(\omega\) is continuous, even, increasing on [0,\(\infty [\), satisfies \(\omega (0)=0\), lim \(\omega\) (t)\(=\infty\), \(\limsup_{t\to \infty}\omega (2t)/\omega (t)0\) with \(\int^{\infty}_{0}(\omega (yt/t^ 2)dt\leq C\omega (y)+C\) for all \(y>0.\) The proof is based on the methods introduced by Carleson and Ehrenpreis, Hörmander's solution of the \({\bar \partial}\)-problem, the Phragmén-Lindelöf principle, and the dual characterization of surjective linear maps between Fréchet-Schwartz-spaces.