Nuclearity of rapidly decreasing ultradifferentiable functions and time-frequency analysis

We use techniques from time-frequency analysis to show that the space $mathcal S_omega$ of rapidly decreasing $omega$-ultradifferentiable functions is nuclear for every weight function $omega(t)=o(t)$ as $t$ tends to infinity. Moreover, we prove that, for a sequence $(M_p)_p$ satisfying the classical condition $(M1)$ of Komatsu, the space of Beurling type $mathcal S_{(M_p)}$ when defined with $L^{2},$norms is nuclear exactly when condition $(M2)$ of Komatsu holds.