Geodesic
Theorems of Ostrovskii type and invariant subspaces of analytic functions
Sbornik. Mathematics, Tome 37 (1980) no. 1, pp. 83-95 
A. Ya. Gil'mutdinova. Theorems of Ostrovskii type and invariant subspaces of analytic functions. Sbornik. Mathematics, Tome 37 (1980) no. 1, pp. 83-95. http://geodesic.mathdoc.fr/item/SM_1980_37_1_a5/
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     title = {Theorems of {Ostrovskii} type and invariant subspaces of analytic functions},
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     volume = {37},
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     url = {http://geodesic.mathdoc.fr/item/SM_1980_37_1_a5/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Suppose that $G$ is a convex domain in $\mathbf C$, $H$ the space of functions holomorphic in $G$ endowed with the topology of uniform convergence on compact sets, and $W$ a closed subspace in $H$ invariant with respect to the operator of differentiation and admitting spectral synthesis. In this paper it is shown that an arbitrary function $f\in W$ may be uniformly approximated by linear combinations of exponential monomials from $W$, not only within $G$ but also in the whole domain of existence of $f$, if the annihilator submodule $I$ of $W$ contains an entire function $\varphi$ of exponential type which on a sequence of circles $|z|=\rho_k$, $\rho_k\uparrow\infty$ as $k\to\infty$, admits the estimate $\ln|\varphi(z)|\leqslant o(|z|)$ ($|z|=\rho_k$, $k\to\infty$). Bibliography: 10 titles.

[1] I. F. Krasichkov-Ternovskii, “Invariantnye podprostranstva analiticheskikh funktsii. I. Spektralnyi sintez na vypuklykh oblastyakh”, Matem. sb., 87(129) (1972), 459–489

[2] I. F. Krasichkov-Ternovskii, “Invariantnye podprostranstva analiticheskikh funktsii. II. Spektralnyi sintez na vypuklykh oblastyakh”, Matem. sb., 88(130) (1972), 3–30

[3] I. F. Krasichkov-Ternovskii, “Invariantnye podprostranstva analiticheskikh funktsii. III. O rasprostranenii spektralnogo sinteza”, Matem. sb., 88(130) (1972), 331–352

[4] A. F. Leontev, Ryady polinomov Dirikhle i ikh obobscheniya, Trudy Matem. in-ta im. V. A. Steklova, XXXIX, 1951 | MR | Zbl

[5] B. Ya. Levin, Raspredelenie kornei tselykh funktsii, Gostekhizdat, Moskva, 1956

[6] G. Polya, “Eine Verallgemeinerung des Fabryschen Lückensatzes”, Nachr. Acad. Wiss. Gottingen Math.-Phys. Klasse, 2 (1927), 187–195

[7] A. Ostrowski, “On the representation of analytic functions by power series”, J. London Math. Soc., 1 (1926), 251–263 | DOI | Zbl

[8] M. G. Valiron, “Sur les solutions des equations differential les lineires d'ordre infini et á coefficients constants”, Ann. scient. Ecole norm, super., 46:1 (1929), 25–53 | MR | Zbl

[9] G. M. Goluzin, Geometricheskaya teoriya funktsii kompleksnogo peremennogo, Gostekhizdat, 1952 | MR

[10] A. F. Leontev, “O predstavlenii funktsii ryadami polinomov Dirikhle”, Matem. sb., 70(112) (1966), 132–144 | MR