A version of the Malliavin--Rubel Theorem on entire functions of exponential type with zeros near the imaginary axis

A. E. Salimova

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 8 (2022), pp. 46-55

Résumé

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

Détail

Keywords: entire function, distribution of zeros of entire function, logarithmic characteristics and measures, Blaschke condition, Redheffer density.