On the Distribution of Zero Sets of Holomorphic Functions. III. Conversion Theorems

Abstract

Let $D$ be a domain in the complex plane $\mathbb C$. It follows from first part of our work that if a non-zero holomorphic function $f$ on $D$ vanishes on a sequence ${\sf Z}\subset D$ and satisfies $|f|\leq M$ on $D$, where $M$ is a subharmonic function on $D$, then the the distribution of ${\sf Z}$ is subordinated to the Riesz measure $\nu_M$ of $M$ in a certain sense. Here we show that this result is "almost reversible".