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Topological properties of convolutor spaces via the short-time Fourier transform. (English) Zbl 1468.46004

Summary: 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.

MSC:

46A13 Spaces defined by inductive or projective limits (LB, LF, etc.)
46E10 Topological linear spaces of continuous, differentiable or analytic functions
46F05 Topological linear spaces of test functions, distributions and ultradistributions
46M18 Homological methods in functional analysis (exact sequences, right inverses, lifting, etc.)
81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
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