Dual approach to certain questions for weighted spaces of holomorphic functions (Q2782548)

Let \(H(G;M)\) be the space of all holomorphic functions \(f\) on a domain \(G\subset\mathbb{C}\) such that NEWLINE\[NEWLINE\log\bigl|f(z)\bigr|\leq M(z)+O(1)NEWLINE\]NEWLINE with \(M(z)\) a continuous function on \(G\). The following problems are under consideration: (1) nontriviality of \(H(G;M)\); (2) description of zero sets; (3) description of sets of uniquenss; (4) representation of meromorphic functions. The problems are shown to be equivalent to certain dual problems for Jensen/reproducing measures on \(G\). This is illustrated by the case of \(G\) the unit disk and \(M(z)\equiv \text{const}\), where the problems dual to (2) and (4) can be easily solved and give classical Nevanlinna theorems.NEWLINENEWLINEFor the entire collection see [Zbl 0980.00031].