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

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 $\mathsf{Z}\subset D$ and satisfies $\mathrm{\mid }f\mathrm{\mid }\le M$ on $D$, where $M$ is a subharmonic function on $D$, then the the distribution of $\mathsf{Z}$ is subordinated to the Riesz measure ${\nu }_M$ of $M$ in a certain sense. Here we show that this result is “almost reversible”.