具有给定零点的指数型小整函数的Malliavin-Rubel定理的推广和发展
Generalization and development of the Malliavin-Rubel theorem on small entire functions of exponential type with given zeros
摘要：
以前，我们开发的技术balayage措施或收费和（$δ$ -）次谐波函数的有限阶到一个封闭的系统的射线$S$与顶点在零上的复平面$\mathbb C$。本文仅利用测度与荷的两种平衡，以及$1$阶下有限型次调和函数的平衡及其差。首先，它是四个闭合射线系统上的属$q=0$的经典平衡：正和负，实和虚半轴$\mathbb R^+$，$-\mathbb R^+$，$i\mathbb R^+$，$-i\mathbb R$。其次，它是从开放的右半平面和左半平面$\mathbb C_{\rm rh}$和$\mathbb C_{\rm lh}$到虚轴$i\mathbb R$上的亏格$q=1$的双侧平衡。经典的Malliavin-Rubel定理给出了指数型整函数存在的充分必要条件（我们写作e.f.e.t.）$f\not\equiv 0$，在给定的正序列${\sf Z}=\{{\sf z}_k\}_{k\in \mathbb N}\subset \mathbb R^+$上消失并且满足对$i\mathbb R$的约束$|f|\leq |g|$，其中$g$是e.f.e.t.，在正序列${\sf W}=\{{\sf w}_k\}_{k\in \mathbb N}\subset \mathbb R^+$上消失。结合这些特殊的平衡过程的属$q=0$和$q=1$允许我们扩展Malliavin-Rubel定理到任意复杂的序列${\sf Z}=\{{\sf z}_k\}_{k\in \mathbb N}\subset \mathbb C$由一对垂直角从虚轴$i\mathbb R$分离，更一般的限制$\ln |f|\leq M$的虚轴$i\mathbb R$，其中$M$是一个次调和函数的有限型下的顺序$1$。
英文摘要：
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 $1$.
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