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

Debrouwere, Andreas; Vindas, Jasson

doi:10.1090/tran/8080

Topological properties of convolutor spaces via the short-time Fourier transform
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by Andreas Debrouwere and Jasson Vindas PDF

Trans. Amer. Math. Soc. 374 (2021), 829-861 Request permission

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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