On nontrivial solutions of the homogeneous Abel problem

Yu. A. Kaz'min

Sbornik. Mathematics, Tome 37 (1980) no. 2, pp. 227-244

Résumé

Let $K$ denote the set of all entire functions $F(z)$ of finite exponential type with the following growth characteristic along the imaginary axis: $$ F(iy)=O(|y|^Ne^{\frac\pi2|y|}),\qquad y\to\infty\quad(N\geqslant0). $$ It is shown in this paper that the general solution of the symmetric Abel interpolation problem $$ F^{(n)}(\pm n)=0,\qquad n=0,1,2,\dots, $$ in the class $K$ is of the form $F(z)=C\sin(\pi z/2)$, where $C$ is an arbitrary constant. Bibliography: 10 titles.