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

TL;DR

This paper investigates the topological and structural properties of weighted convolutor spaces, extending classical results and providing new proofs for the completeness of certain function spaces.

Contribution

It characterizes weight sequences for which convolutor spaces are ultrabornological and offers the first direct proof of the completeness of the space of very slowly increasing smooth functions.

Findings

Characterization of weight sequences for ultrabornological convolutor spaces

Generalization of Grothendieck's classical result

First direct proof of completeness of  of very slowly increasing functions

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.