Interpolation problems, nontrivial expansions of zero, and representing systems

Yu. F. Korobeinik

Izvestiya. Mathematics , Tome 17 (1981) no. 2, pp. 299-337

Résumé

Let $G$ be a convex domain with support function $h(-\varphi)$, and let $\{\lambda_k\}$ be distinct complex numbers. In this paper the author determines when the system $\{e^{\lambda_kz}\}$ is absolutely representing in the space $A(G)$ of functions analytic in $G$, with the topology of uniform convergence on compact sets. In particular he proves the Theorem. {\it Let $L(\lambda)$ be an exponential function with indicator $h(\varphi)$ and simple zeros $\{\lambda_n\}_{n=1}^\infty$. For the system $\{e^{\lambda_kz}\}_{k=1}^\infty$ to be absolutely representing in $A(G)$ it is necessary and sufficient that either of the following two conditions hold}: 1) {\it The system $\{e^{\lambda_kz}\}_{k=1}^\infty$ has a nontrivial expansion of zero in $A(G)$, i.e. $\sum_{n=1}^\infty b_ne^{\lambda_nz}=0$ for every $z\in G$}. \smallskip 2) $L(\lambda)$ is a function of completely regular growth and there exists a function $C(\lambda)$ of class $[1,0]$ such that $$ \varlimsup_{n\to\infty}\left[\frac1{|\lambda_n|}\ln\left|\frac{C(\lambda_n)}{L^{'}(\lambda_n)}\right|+h(\arg\lambda_n)\right]\leqslant0. $$ Bibliography: 16 titles.