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Abstract

Let $f$ be a holomorphic function of Carleman type in a bounded convex domain $D$ of the plane. We show that $f$ can be expanded in a series $f=\sum _n f_{n}$, where $f_{n}$ is a holomorphic function in $D_{n}$ satisfying $\mathop {\rm sup}_{z\in D_{n}}|f_{n}(z)| \leq C\varrho ^{n}$ for some constants $C>0$ and $0<\varrho <1$, and where $(D_{n})_{n}$ is a suitably chosen sequence of decreasing neighborhoods of the closure of $D$. Conversely, if $f$ admits such an expansion then $f$ is of Carleman type. The decrease of the sequence $D_n$ characterizes the smoothness of $f$.