Weakly regular sets in the space of functions of finite order in the half-plane

In this article, the concept of a weakly regular set in the space of analytical functions of the finite order greater than unity in the upper half-plane of complex variable is introduced. The sequence $A=\{a_n=r_n e^{i \theta_n}, n=1,2, \ldots \}, \, A \subset C_+$, is called weakly regular in $$C_+$ by order $\rho > 1$ if one of the following conditions $(C'_+)$ or $(C_+)$ is satisfied: $(C_+)$ 1) Among the points of the set $A$ there are no multiples; 2) for any $\epsilon>0$ there exists $R= R(\epsilon)$ such that for $r_n> R$ the disks of radius $d_n(\epsilon)=sin^{\frac{1}{2}}\theta_n r_n^{1- \frac{\rho+\epsilon}{2}}$ with centers $a_n$ do not intersect; 3) for any $\epsilon>0 sum^\infty_{n=1} \frac{sin \theta_n}{r_n^{\rho+\epsilon}<\infty (C'_+)$ 1') Among the points of the set $A$ there are no multiples and there are no points with the same modules; 2') conditions 1) and 3) are true; 3') for any $\epsilon>0$ there exists $R= R(\epsilon)$ such that for $r_n>r_k>R$ the inequality $|a_n|geq |a_k|+\frac{Im a_k}{|a_k|^{\rho+\epsilon}}$ are true. It is proved that such sets are interpolation in the sense of free interpolation in this space.