The problem of multiple interpolation in the~half-plane in the~class of analytic functions of finite order and normal type

K. G. Malyutin

Sbornik. Mathematics, Tome 78 (1994) no. 1, pp. 253-266

Résumé

The problem of multiple interpolation is considered in the class $[\rho(r),\infty)^+$ of functions of at most normal type for the proximate order $\rho(r)$ in the upper half-plane $C^+\colon f^{(k-1)}(a_n)=b_{n,k}$, $k = 1,\dots,q_n$, $n=1,2,\dots$, where the divisor $D=\{a_n,\,q_n\}$ has limit points only on the real axis, and the numbers $\{b_{n,k}\}$ satisfy the condition $$ \varlimsup_{n\to\infty}r_n^{-\rho(r_n)}\ln\sup_{1\leqslant k\leqslant q_n}\dfrac{(\Lambda_n)^{k-1}|b_{n,k}|}{(k-1)!}\infty. $$ The following result is valid. Theorem. {\it $D$ is an interpolation divisor in the class $[\rho(r),\infty)^+$ if and only if $$ \varlimsup_{n\to\infty}r_n^{-\rho(r_n)}\ln\frac{q_n!}{|E^{(q_n)}(a_n)|(\Lambda _n)^k}\infty, $$ where $E(z)$ is the canonical product of the set $D$}. Necessary and sufficient conditions are also found in terms of the measure determined by the divisor $D$: $\mu(G)=\sum_{a_n\in G}q_n\sin(\arg a_n)$.