Geodesic
On nontrivial solutions of the homogeneous Abel problem
Sbornik. Mathematics, Tome 37 (1980) no. 2, pp. 227-244 
Yu. A. Kaz'min. On nontrivial solutions of the homogeneous Abel problem. Sbornik. Mathematics, Tome 37 (1980) no. 2, pp. 227-244. http://geodesic.mathdoc.fr/item/SM_1980_37_2_a4/
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     author = {Yu. A. Kaz'min},
     title = {On nontrivial solutions of the homogeneous {Abel} problem},
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     volume = {37},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1980_37_2_a4/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] G. Polya, “Prolongement analytique”, Enseignment Math., 22 (1922), 298–299

[2] G. Valiron, “Sur la formule d'interpolation de Lagrange”, Bull. Sci. Math., Sér. 2, 49 (1925), 181–192 ; 203–224 | Zbl

[3] R. P. Boas, Entire functions, Academ. Press, New York, 1954 | MR | Zbl

[4] P. Malliavin, L. A. Rubel, “On small entire functions of exponential type with given zeros”, Bull. Soc. Math. France, 89 (1961), 175–206 | MR | Zbl

[5] Yu. A. Kaz'min, “On a theorem of G. Polya”, Analysis Math., 2 (1976), 99–116 | DOI | MR

[6] Yu. A. Kazmin, “K interpolyatsionnoi zadache Abelya”, Matem. zametki, 11:4 (1972), 353–364 | MR

[7] Yu. A. Kazmin, “Zamykaniya lineinoi obolochki odnoi sistemy funktsii”, Sib. matem. zh., XVIII:4 (1977), 799–805 | MR

[8] N. I. Muskhelishvili, Singulyarnye integralnye uravneniya, Fizmatgiz, Moskva, 1962

[9] F. D. Gakhov, Kraevye zadachi, izd-vo “Nauka”, Moskva, 1977 | MR | Zbl

[10] G. Pólya, “Beitrag zur Verallgemeinerung des Verzerrungsatz auf Mehrfachzusammenhangender Gebiete. III”, Sitzungsber. preuss. Akad., 1929, 55–62 ; Г. М. Голузин, Геометрическая теория функций комплексного переменного, Гостехиздат, Москва–Ленинград, 1952, 333–337 | Zbl | MR