The distribution of zeros of entire functions of exponential type with restrictions on growth along the imaginary axis

TL;DR

This paper establishes a criterion for the existence of entire functions of exponential type with prescribed zeros and growth restrictions along the imaginary axis, extending previous results and exploring various applications.

Contribution

It generalizes and develops joint results of Malliavin and Rubel regarding entire functions with growth constraints and prescribed zeros.

Findings

Provides a new criterion for such entire functions' existence.

Extends previous theorems to broader classes of functions.

Applications include multipliers, analytic functionals, and completeness of exponential systems.

Abstract

Let $g$ be a entire function of exponential type on the complex plane $\mathbb C$, $Z=\{ z_k\}_{k=1,2,\dots}$ be a sequence of points in $\mathbb C$. We give a criterion for the existence of an entire function $f\neq 0$ of exponential type that vanishes on $ Z$ and satisfies the constraint $\ln |f(iy)|\leq \ln |g(iy)|+o(|y|)$, $y\to \pm\infty$. Our results generalize and develop a joint results of P. Malliavin and L. A. Rubel. Applications to multipliers for entire functions of exponential type, to analytic functionals and their convolutions on the complex plane, as well as to the completeness of exponential systems in the spaces of locally analytic functions on compacts in terms of the width of these compacts are given.