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Projective representations of spaces of quasianalytic functionals. (English) Zbl 1062.46022

In [Trans. Am. Math. Soc. 272, 107–160 (1982; Zbl 0599.46026)], K. D. Bierstedt, R. Meise and W. H. Summers considered weighted (LF)-spaces \({\mathcal V}H(G)\) of holomorphic functions on an open set \(G \subset \mathbb{C}^N\) and introduced projective hulls \(H\overline{V}(G)\) of these spaces. \({\mathcal V}H(G)\) is always continuously embedded in \(H\overline{V}(G)\), and \(H\overline{V}(G)\) is a projective limit of Banach spaces. The question whether the (LF)-space coincides algebraically with its projective hull (which is always the case for weighted (LB)-spaces) and whether it carries the topology induced from its projective hull was called the question of projective description or representation.
Let \({\mathcal A}(G)\) denote the space of all complex valued real-analytic functions on an open convex subset \(G\) of \(\mathbb{R}^N\). Its strong dual \({\mathcal A}(G)'_b\) is isomorphic to an (LF)-space \({\mathcal FA}'(G)\) of entire functions on \(\mathbb{C}^N\) via the Fourier-Laplace transform. Each step space of this (LF)-space is a Fréchet space whose topology is given by weighted sup-seminorms. L. Ehrenpreis [Am. J. Math. 82, 522–588 (1960; Zbl 0098.08401)] showed that the topology of \({\mathcal FA}'(\mathbb{R}^N)\) cannot be described by weighted sup-seminorms. J. Bonet and R. Meise [Math. Scand. 94, No. 2, 249–266 (2004; Zbl 1064.46018)] proved that \({\mathcal FA}'(G)\) has a strictly finer topology than its projective hull.
In the present article, the authors prove, however, that \({\mathcal FA}'(G)\) coincides with its projective hull algebraically. The authors also extend this result to the spaces \({\mathcal E}'_{\{\omega\}}(G)\) of quasianalytic functionals, where \({\mathcal E}_{\{\omega\}}(G)\) denotes the space of all \(\omega\)-ultradifferentiable functions of Roumieu type on \(G\) for a given (quasianalytic) weight function \(\omega\). More precisely, they prove that the (LF)-space \({\mathcal FE}'_{\{\omega\}}(G)\) of entire functions on \(\mathbb{C}^N\) which arise as the Fourier-Laplace transforms of the elements of \({\mathcal E}_{\{\omega\}}(G)'_b\) coincides algebraically with its projective hull and has a strictly finer topology. (The latter assertion already follows from the above quoted article of Bonet and Meise.) The corresponding result for non-quasianalytic classes \({\mathcal E}_{\{\omega\}}(G)\) was obtained in [J. Bonet and R. Meise, J. Math. Anal. Appl. 255, 122–136 (2001; Zbl 0960.46027)]. The proof in the present case is different from the one in the non-quasianalytic case.

MSC:

46E10 Topological linear spaces of continuous, differentiable or analytic functions
46A13 Spaces defined by inductive or projective limits (LB, LF, etc.)
46F05 Topological linear spaces of test functions, distributions and ultradistributions
46F15 Hyperfunctions, analytic functionals
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