On the reconstruction of a function from the known coefficients of the corresponding Dirichlet series

Let $L(\lambda)=\displaystyle\sum_{k=0}^\infty c_k\lambda^k$ be an entire function of order $\rho_1$ ($1<\rho_1<2$). We denote by $\lambda_1,\lambda_2,\dots,\lambda_n,\dots$ the zeros of the function $L(\lambda)$. It is assumed that all the zeros of the function $L(\lambda)$ are simple, and that $\lim_{n\to\infty}\frac n{\lambda_n^{\rho_1}}=\tau\ne0,\infty$.
We take an arbitrary function $F(z)=\sum_{n=0}^\infty b_nz^n$ of order $\nu<\frac{\rho_1}{\rho_1-1}$. We associate with the function $F(z)$ the series
\begin{equation} F(z)\thicksim\sum_{n=1}^\infty A_ne^{\lambda_nz},\qquad A_n=\frac{\omega_L(\lambda_n,F)}{L'(\lambda_n)}, \end{equation}
where
$$ \omega_L(u,\,F)=\sum_{k=1}^\infty c_k[F^{(k-1)}(0)+uF^{(k-2)}(0)+\ldots+u^{k-1}F(0)]. $$
The series (1) is, in general, divergent. In particular, the series (1) can converge absolutely and uniformly throughout the plane, but not to the function $F(z)$. In the present paper a method is indicated for the reconstruction of the function $F(z)$ from the known coefficients $A_n$ ($n=1,2,\dots$) of (1).
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