Recovery of a function from the coefficients of its Dirichlet series

V. V. Napalkov

Matematičeskie zametki, Tome 17 (1975) no. 4, pp. 545-553

Résumé

Let $L(\lambda)$ be an entire function of exponential type, let $\gamma(t)$ be the function associated with $L(\lambda)$ in the sense of Borel, let $\overline D$ be the smallest closed convex set containing all the singular points of $\gamma(t)$, let $\lambda_0,\lambda_1,\dots,\lambda_n\dots$ be the simple zeros of $L(\lambda)$, and let A $\overline D$ be the space of functions analytic on $\overline D$ with the topology of the inductive limit. With an arbitrary $f(z)\in A(\overline D)$ we can associate the series \begin{gather*} f(z)\sim\sum_{n=0}^\infty a_ne^{\lambda_nz},\quad a_n=\frac{\omega_L(\lambda_n,f)}{L'(\lambda_n)}, \\ \omega_L(\mu,f)=\frac1{2\pi i}\int_\mathscr C\gamma(t)\int_0^tF(t-\eta)e^{\mu\eta}\,d\eta\,dt, \end{gather*} where $\mathscr C$ is a closed contour containing $\overline D$ , on and inside of which $f(z)$ is analytic. We give a method of recovering $f(z)$ from the Dirichlet coefficients $a_n$.