A Version of the Malliavin–Rubel Theorem on Entire Functions of Exponential Type with Zeros near the Imaginary Axis

Abstract—

Let $Z$ and $W$ be two distributions of points on the complex plane $\mathbb{C}$. In the case of $Z$ and $W$, lying on the positive half-line ${{\mathbb{R}}^{ + }} \subset \mathbb{C}$, the classic Malliavin–Rubel theorem from the 1960s, gives a necessary and sufficient correlation between $Z$ and $W$, when for each entire function $g \ne 0$ exponential type that vanishes on $W$, there exists an entire function $f \ne 0$ exponential type that vanishes on $Z$, with the constraint ${\text{|}}f{\text{|}} \leqslant {\text{|}}g{\text{|}}$ on the imaginary axis $i\mathbb{R}$. In subsequent years, this theorem was extended to $Z$ and $W$, located outside of some pair of angles containing $i\mathbb{R}$ inside. Our version of the Malliavin–Rubel theorem admits the location of $Z$ and $W$ near and on $i\mathbb{R}$.