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.
Nuclearity of rapidly decreasing ultradifferentiable functions and time-frequency analysis
Alessandro Oliaro;
2021-01-01
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 $(M2)'$ of Komatsu holds.
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