Theorems of Ostrovskii type and invariant subspaces of analytic functions

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 $\ln|\varphi(z)|\leqslant o(|z|)$ ($|z|=\rho_k$, $k\to\infty$).
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