  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.
