(DFS)-spaces of holomorphic functions invariant under differentiation
Let \(G \subset \mathbb{C}\) be a bounded convex domain and let \(d_G(z) = \text{ inf} \{| z-t| ;\;t \in \partial G\}\), \(z \in G\). Let \(A^{-n}(G)\) denote the Banach space of all holomorphic functions on \(G\) for which \(| | f| | _n = \text{ sup}_G | f| d_G^n < \infty\), and \(A^{-\infty}(G) = \text{ ind}_n A^{-n}(G)\) be the space of holomorphic functions of polynomial growth on \(G\), a (DFS)-space on which differentiation is a continuous operation. The author shows that the Laplace transform is a topological isomorphism of the strong dual \(A^{-\infty}(G)'_b\) onto the Fréchet space \(A_G^{-\infty}\) of all entire functions \(f\) for which \[ | f| _n = \text{ sup}_{z \in \mathbb{C}}\;\frac{| f(z)| (1+| z| )^n}{\text{ exp} H_G(z)} < +\infty \] for each \(n \in \mathbb{N}\), where \(H_G\) denotes the support function of \(G\), \(H_G(z) = {\sup}_{t \in G} \text{ Re} (tz)\), \(z \in \mathbb{C}\). The following, however, is the main result of the article under review: Let \(A^{\infty}(G)\) denote the Fréchet space of all \(C^{\infty}\)-functions on \(\overline{G}\) which are holomorphic on \(G\). Then \(A^{-\infty}(G)\) is the unique (DFS)-space \(H\) which contains \(A^{\infty}(G)\), is continuously embedded in \(A^{-\infty}(G)\) and invariant under differentiation.
holomorphic functions of polynomial growth
duality of spaces of holomorphic functions
weighted Fréchet space of entire functions
scientific article; zbMATH DE number 2107208
scientific article