On conditions for representability of entire functions by certain general series

Let $f(z)$ be an entire function of order $\rho$ and $L(\lambda)$ an entire function of order $\rho_1>\rho$ with simple zeros $\lambda_1,\dots,\lambda_k,\dots$ . A series $\sum_1^\infty\alpha_kf(\lambda_kz)$ is assigned (according to a specific rule) to an arbitrary entire function $F(z)$ of order $\nu<\frac{\rho\rho_1}{\rho_1-\rho}$. Necessary and sufficient conditions on $L(\lambda)$ are found under which this series always converges to $F(z)$ in some topology.
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