Balayage on a system of rays and entire functions of completely regular growth

This paper presents a technique for constructing functions that are subharmonic in the complex plane, agree with a given subharmonic function $u$ on a system $S$ of rays with vertex at the origin, and are harmonic outside $S$. For a wide class of systems $S$, this technique permits one to obtain criteria for the complete regularity of growth of entire functions $f$ on $S$ in terms of the balayage of the distribution of zeros of $f$.