On~the representation of analytic functions in an open region by Dirichlet series

A. F. Leont'ev

Sbornik. Mathematics, Tome 10 (1970) no. 4, pp. 503-530

Résumé

In the author's paper (On the representation of analytic functions by Dirichlet series, Mat. Sb. (N.S.), 80(122) (1969), 117–156) a theorem was proved stating that every function $f(z)$, analytic in a finite convex region $D$ and continuous in $\overline D$, can be represented in $D$ by a Dirichlet series. Here we have obtained a definitive result: any function $F(z)$, analytic in $D$, is representable in $D$ by a Dirichlet series. The proof is based on the following assertion. Let $F(z)$ be a function analytic in a finite convex region $D$. There exist a function $f(z)$, analytic in $D$ and continuous in $\overline D$, and an operator $M(y)=\sum_0^\infty c_ny^{(n)}(z)$ with characteristic function $L(\lambda)=\sum_0^\infty c_n\lambda^n$ from the class $[1,0]$, such that $M(f)=F(z)$. Bibliography: 4 titles.