Distribution of Zeros of Exponential-Type Entire Functions with Constraints on Growth along a Line

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