Holomorphic series expansion of functions of Carleman type

Taib Belghiti¹

¹ Department of Mathematics Faculty of Sciences B.P. 133, Kénitra, 14000, Morocco

Annales Polonici Mathematici, Tome 84 (2004) no. 3, pp. 219-224.

Résumé

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 $f$.

Détail

DOI : 10.4064/ap84-3-4

Keywords: holomorphic function carleman type bounded convex domain plane expanded series sum where holomorphic function satisfying mathop sup leq varrho constants varrho where suitably chosen sequence decreasing neighborhoods closure conversely admits expansion carleman type decrease sequence characterizes smoothness