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arXiv · 1801.09246

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

Andreas Debrouwere·Jasson Vindas
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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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Andreas Debrouwere, Jasson Vindas. 2018-01-28. Topological properties of convolutor spaces via the short-time Fourier transform. https://doi.org/10.1090/tran/8080

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