# Nuclearity of rapidly decreasing ultradifferentiable functions and time-frequency analysis

## TL;DR

This paper demonstrates the nuclearity of certain ultradifferentiable function spaces using time-frequency analysis, establishing conditions under which these spaces are nuclear based on weight functions and Komatsu's conditions.

## Contribution

It introduces new nuclearity results for ultradifferentiable function spaces using time-frequency analysis techniques, linking nuclearity to specific growth conditions.

## Key findings

- The space $\
- $ ext{S}_ ext{ω}$ is nuclear for all weight functions with $ω(t)=o(t)$.
- The space $ ext{S}_{(M_p)}$ is nuclear if and only if condition $(M2)'$ holds.

## 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.