论文标题
Malliavin-Rubel定理的概括和开发在指数类型的小函数上具有给定零
Generalization and development of the Malliavin-Rubel theorem on small entire functions of exponential type with given zeros
论文作者
论文摘要
以前，我们开发了措施或费用的Balayage的技术，以及（$δ$ - ）有限订单的次谐波功能，在复杂平面$ \ Mathbb c $上的零顶点的封闭系统的封闭系统上。在本文中，我们仅使用两种量度和费用的Balayage，以及有限类型的次谐波功能，在$ 1 $及其差异下。首先，它是四个封闭射线系统上的属$ q = 0 $的经典balayage：正面和负面，以及真实和虚构的半轴$ \ Mathbb r^+$，$ - \ Mathbb r^+$，$ i \ Mathbb r^+$，$ - i \ $ -i \ Mathbb r $。其次，它是从左右和左半平台$ \ mathbb c _ {\ rm rh} $和$ \ mathbb c _ {\ rm lh} $上的双面balayage $ q = 1 $。经典的malliavin-rubel定理给出了指数类型的全部功能的必要条件（我们编写e.f.e.t.）$ f \ not \ equiv 0 $，在给定的正序$ {\ sf z} = \} = \ {{\ sf z} {\ sf z} _k \} _ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ac { \ Mathbb r^+$并满足约束$ | f | \ leq | g | $ on $ i \ mathbb r $，其中$ g $是e.f.e.t.，在正序$ {\ sf w} = \ \ \ \ \ \ {\ sf w} _k \} _k \ \} _ { r^+$。这些特殊的Balayage过程的组合$ q = 0 $和$ q = 1 $使我们能够将Malliavin-Rubel定理扩展到任意复杂序列$ {\ sf z} = \ {{\ sf z} {\ sf z} _k} _k \} _k \} _ {假想轴$ i \ mathbb r $，具有更多的一般限制$ \ ln | f | \ leq m $在虚构轴$ i \ mathbb r $上，其中$ m $是订单$ 1 $下的有限型类型的次谐波功能。
Previously, we developed the technique of balayage of measures or charges and ($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 $1$.
