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

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 $M(f)=F(z)$.
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