On~the representation of analytic functions by Dirichlet series

A. F. Leont'ev

Sbornik. Mathematics, Tome 9 (1969) no. 1, pp. 111-150

Résumé

We have earlier proved (RZhMat., 1966, 2B149, 11B94) a theorem on the representation of an arbitrary function analytic in a closed convex region $\overline D$ by a Dirichlet series in the open region $D$. In this paper we prove that any function analytic in an open convex finite region $D$ and continuous in $\overline D$ can be represented by a Dirichlet series with coefficients which can be computed by means of specific already-known formulas. We also prove that if the convex region $D$ is bounded by a regular analytic curve, then any function analytic in $D$ can be expanded in a Dirichlet series in $D$. These two theorems are based on the following theorem from the theory of entire functions. Let $D$ be a finite open region, $K(\theta)$ the support function of $D$, $h(\theta)=H(-\theta)$, and $\varphi(r)$ a function satisfying the conditions $$ 0\varphi(r)\uparrow\infty,\qquad\lim_{r\to\infty}\frac{\ln\varphi(r)} r=0. $$ Then there exists an entire function $L(\lambda)$ of exponential type with growth indicator $h(\theta)$ and completely regular growth, which satisfies the following conditions: 1) All the zeros $\lambda_1,\lambda_2,\dots$ of $L(\lambda)$ are simple, and $|\lambda_{n+1}|-|\lambda_n|\geqslant h>0$. 2) We have the estimate $$ \bigl|L(re^{i\theta})\bigr|\frac{e^{h(\theta)r}}{\varphi(r)},\qquad r>r_0. $$ 3) The sequence $\{\lambda_n\}$ is part of a sequence $\{\mu_n\}$, $\lim_{n\to\infty}\frac n{|\mu_n|}\infty$, which depends on the region $D$ but not on the function $\varphi(r)$. In this paper we prove an analogous theorem for entire functions of arbitrary finite order $\rho$. Bibliography: 6 titles.