Entire Functions of Sine Type for Convex Apeirogons

Abstract

The paper defines a class of convex apeirogons $D$ on the complex plane for which there exists an entire function $L$ that has the properties of the generating function of basis systems of exponentials in Hilbert spaces of functions analytic on the domain $D$. Namely, if $h_{j}\rightarrow 0$ is the sequence of side’s lengths of a convex apeirogon $D$ and $H(z)$ is the support function of $D$, then there is an entire function with next properties:

1) if $h(t)=\sum_{h_{j}^{-1}\leq t}1$ is the counting function of the sequence $h_{j}^{-1}$, then

$$|L(z)|\leq\text{Const}\,e^{H(z)+h(|z|)\ln 2},\quad z\in\mathbb{C};$$

2) all zeros of the function $L$ lie on the rays perpendicular to the sides of the apeirogon $D$;

3) if $\lambda_{n}^{j}$ are the zeros lying on the ray perpendicular to the side of length $h_{j}$ and $B(\lambda_{n}^{j},\delta h_{j}^{-1})$ are the disks with center $\lambda_{n}^{j}$ and radius $\delta h_{j}^{-1}$, then

$$|\ln|L(z)||\geq\text{Const}\,e^{H(z)+h(|z|)\ln 2},\quad z\notin\bigcup B(\lambda_{n}^{j},\delta h_{j}^{-1}),$$

for some constant $\delta>0$.