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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Topological properties of convolutor spaces via the short-time Fourier transform


Authors: Andreas Debrouwere and Jasson Vindas
Journal: Trans. Amer. Math. Soc. 374 (2021), 829-861
MSC (2010): Primary 46A13, 46E10, 46F05; Secondary 46M18, 81S30
DOI: https://doi.org/10.1090/tran/8080
Published electronically: November 18, 2020
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Abstract: We discuss the structural and topological properties of a general class of weighted $ L^1$ convolutor spaces. Our theory simultaneously applies to weighted $ \mathcal {D}'_{L^1}$ spaces as well as to convolutor spaces of the Gelfand-Shilov spaces $ \mathcal {K}\{M_p\}$. In particular, we characterize the sequences of weight functions $ (M_p)_{p \in \mathbb{N}}$ for which the space of convolutors of $ \mathcal {K}\{M_p\}$ is ultrabornological, thereby generalizing Grothendieck's classical result for the space $ \mathcal {O}'_{C}$ of rapidly decreasing distributions. Our methods lead to the first direct proof of the completeness of the space $ \mathcal {O}_{C}$ of very slowly increasing smooth functions.


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Additional Information

Andreas Debrouwere
Affiliation: Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, 9000 Gent, Belgium
Email: andreas.debrouwere@ugent.be

Jasson Vindas
Affiliation: Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, 9000 Gent, Belgium
Email: jasson.vindas@ugent.be

DOI: https://doi.org/10.1090/tran/8080
Keywords: Convolutor spaces, short-time Fourier transform, completeness of inductive limits, Gelfand-Shilov spaces
Received by editor(s): September 21, 2018
Received by editor(s) in revised form: June 8, 2019, and November 21, 2019
Published electronically: November 18, 2020
Additional Notes: The first author was supported by FWO-Vlaanderen through the postdoctoral grant 12T0519N
The second author was supported by Ghent University through the BOF-grants 01J11615 and 01J04017.
Article copyright: © Copyright 2020 American Mathematical Society