Conditions for solvability for analytic functions in a Dirichlet series in the half-plane (Q1082471)

Let f(z) be a holomorphic function in the halfplane \(D=\{z\in {\mathbb{C}}:\) Re z\(<0\}\) being continuous in its closure \(\bar D.\) Let further L(\(\lambda)\) be an entire function with simple zeros \(\lambda_ n\), satisfying some a priori assumptions. In the paper conditions are given on L(\(\lambda)\) being necessary and sufficient for the following equalities to hold \[ e^{\lambda z}=\sum^{\infty}_{n=1}\frac{L(\lambda)}{\lambda -\lambda_ n}\frac{e^{\lambda_ nz}}{L'(\lambda_ n)},\quad z\in D,\quad \lambda \in {\mathbb{C}}, \] f(z)\(=\sum^{\infty}_{n=1}A_ ne^{\lambda_ nz}\), \(z\in D\).