Minimal absolutely representing systems of exponentials for \(A^{-\infty}(\varOmega)\) (Q719511)

Let \(\Omega\) be a bounded convex domain in the complex plane \(\mathbb{C}\). The algebra \(A^{-\infty}(\Omega)\) consists of those holomorphic functions \(f\) in \(\Omega\) having no more than polynomial growth near the boundary \(\partial\Omega\), in the sense that for some \(n\in\mathbb{N}\), the product \(f(z)\cdot d(z)^n\) is bounded, where \(d(z)\) denotes the distance from \(z\) to the boundary. The paper is about representing such \(f\) by Dirichlet series \(f(z) = \sum_{k=1}^\infty c_k e^{\lambda_kz}\), absolutely-convergent in \(\Omega\), where the frequencies \(\lambda_k\) are the elements of a fixed sequence \(\Lambda\subset\mathbb{C}\). The sequence of exponentials \((e^{\lambda_kz})\) is denoted \(\mathcal E_\Lambda\), and is called an ARS (absolutely representing system) if each \(f\in A^{-\infty}(\Omega)\) has such a representation. An ARS remains an ARS when any finite number of terms is removed. An ARS is called minimal if it ceases to be an ARS when any infinite subsequence is removed. It is proved that for each \(\Omega\) there exists a minimal ARS. This depends on a result that gives a sufficient condition for an ARS to be minimal. In more detail, a dual space \(A^{+\infty}_{\Omega}\) (consisting of certain entire functions) is defined, and the authors focus on frequency sets \(\Lambda\) that are subsets of the zero set of some \(L\in A^{\infty}_{\Omega}\). They say this is the most interesting case. Theorem 1.5 states: Let \(\mathcal E_\Lambda\) be an ARS. If there exists some \(L\in A^{\infty}_{\Omega}\) such that \(\Lambda\subset L^{_1}(0)\), then \(\mathcal E_\Lambda\) is a minimal ARS. This result, in turn, depends on the key result, Theorem 1.2, which gives necessary and sufficient conditions for a \(\Lambda\) that is contained in \(L^{-1}(0)\) for some \(L\in A^{+\infty}_\Omega\) to be an ARS. One necessary and sufficient condition is that \(\mathcal E_\Lambda\) admits an absolutely nontrivial expansion of zero, in the sense that a nontrivial Dirichlet series with these exponentials converges absoltely to zero in \(\Omega\). In contrast to this abstract equivalence, the second necessary and sufficient condition is rather complicated and technical, involving a couple of growth conditions in which appear the supporting function of \(\Omega\): \[ H_\omega(\lambda):= \sup_{z\in\Omega} \mathrm{Re} z\lambda, \text{ for all }\lambda\in\mathbb{C}, \] and the condition that \(L^{-1}(0)\setminus \Lambda\) be finite.