Nuclearity of rapidly decreasing ultradifferentiable functions and time-frequency analysis

Abstract

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 $(M2{)}^{\prime }$ of Komatsu holds.