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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Acknowledgements
We thank the reviewer very much for the careful reading of our manuscript and the comments to improve the paper. The first three authors were partially supported by the Project FFABR 2017 (MIUR), and by the Projects FIR 2018 and FAR 2018 (University of Ferrara). The first and third authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The research of the second author was partially supported by the project MTM2016-76647-P and the grant BEST/2019/172 from Generalitat Valenciana. The fourth author is supported by FWF-project J 3948-N35.
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Boiti, C., Jornet, D., Oliaro, A. et al. Nuclearity of rapidly decreasing ultradifferentiable functions and time-frequency analysis. Collect. Math. 72, 423–442 (2021). https://doi.org/10.1007/s13348-020-00296-0
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DOI: https://doi.org/10.1007/s13348-020-00296-0
Keywords
- Nuclear spaces
- Weighted spaces of ultradifferentiable functions of Beurling type
- Gabor frames
- Time-frequency analysis