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
MathSciNet review:
4196379
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Abstract | References | Similar Articles | Additional Information
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
MR Author ID:
1154620
Email:
andreas.debrouwere@ugent.be
Jasson Vindas
Affiliation:
Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, 9000 Gent, Belgium
MR Author ID:
795097
ORCID:
0000-0002-3789-8577
Email:
jasson.vindas@ugent.be
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