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

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