A duality theorem between spaces of holomorphic functions of exponential growth
Let \(N\) be a nuclear Fréchet space whose topology is given by an increasing sequence of Hilbertian norms \(\{|\cdot|_p:p\in \mathbb{N}\}\). Let \(N_p\) denote the completion of \(N\) with the norm \(|\cdot|_p\) and let \(N_{-p}\) stand for the conjugate space of \(N_p\). \(N'\) is the topological dual of \(N\). Let \(\lambda\) be a Young function. If \((B,\|\cdot\|)\) is a Banach space, \(m>0\) and \(H(B)\) is the linear space of the entire functions on \(B\) the following Banach space is defined \[ \text{Exp}(B, \lambda, m):= \{f\in H(B):\|f\|_{\lambda, m}:= \sup\{|f(z)|e^{- \lambda(m\|z\|)}: z\in B\}< \infty\}. \] Let \(\theta\) be a Young function and let \(\theta^*\) be its conjugate. The following spaces are defined \[ {\mathcal F}_\theta(N'):= \bigcap_{p\geq 0,m>0} \text{Exp}(N_{-p}, \theta,m) \] with the projective limit topology and \[ {\mathcal G}_{\theta^*}(N):= \bigcup_{p\geq 0,m>0} \text{Exp}(N_p, \theta^*,m) \] with the inductive limit topology. \({\mathcal T}_\theta(N')^*\) is the strong dual of \({\mathcal F}_\theta(N')\). In this paper the Laplace transform \({\mathcal L}\) on \({\mathcal F}_\theta(N')^*\) is defined and it is shown that \({\mathcal L}\) defines a topological isomorphism from \({\mathcal F}_\theta(N')^*\) onto \({\mathcal G}_{\theta^*}(N)\).
spaces of holomorphic functions of exponential growth
scientific article; zbMATH DE number 1423449