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.

中文翻译：

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快速下降的超微分函数的核性和时频分析

我们使用来自时频分析的技术来证明，迅速减小的\（\ omega \）-超能函数的空间\（{\ mathcal {S}} _ \ omega \）对于每个权重函数\（\ omega（ t）= o（t）\），因为t趋于无穷大。此外，我们证明，对于满足小松经典条件（M 1）的序列\（（M_p）_p \），Beurling类型\（{\ mathcal {S}} _ {{M_p）} \）的空间当用小松的条件\（（M2）'\）成立时，用\（L ^ {2} \）范数定义时，正是核的。