Generalization and development of the Malliavin-Rubel theorem on small entire functions of exponential type with given zeros

Abstract

Previously, we developed the technique of balayage of measures or charges and ($δ$-)subharmonic functions of finite order onto an closed system of rays $S$ with a vertex at zero on the complex plane $\mathbb C$. In this article, we use only two kinds of balayage of measure and charge, as well as of subharmonic functions of finite type under the order $1$ and their differences. First, it is a classical balayage of the genus $q=0$ on a system of four closed rays: positive and negative, and real and imaginary semi-axis $\mathbb R^+$, $-\mathbb R^+$, $i\mathbb R^+$, $-i\mathbb R$. Second, it is two-sided balayage of genus $q=1$ from the open right and left half-planes $\mathbb C_{\rm rh}$ and $\mathbb C_{\rm lh}$ onto the imaginary axis $i\mathbb R$. The classical Malliavin-Rubel theorem gives necessary and sufficient conditions of the existence of an entire function of exponential type (we write e.f.e.t.) $f\not\equiv 0$, vanishing on the given positive sequence ${\sf Z}=\{{\sf z}_k\}_{k\in \mathbb N}\subset \mathbb R^+$ and satisfying the constraint $|f|\leq |g|$ on $i\mathbb R$, where $g$ is an e.f.e.t., vanishing on positive sequence ${\sf W}=\{{\sf w}_k\}_{k\in \mathbb N}\subset \mathbb R^+$. A combination of these special balayage processes of genus $q=0$ and $q=1$ allows us to extend the Malliavin-Rubel theorem to arbitrary complex sequences ${\sf Z}=\{{\sf z}_k\}_{k\in \mathbb N}\subset \mathbb C$ separated by a pair of vertical angles from the imaginary axis $i\mathbb R$, with much more general restrictions $\ln |f|\leq M$ on the imaginary axis $i\mathbb R$, where $M$ is an subharmonic function of finite type under the order $1$.