Geodesic
A note on formal power series
Commentationes Mathematicae Universitatis Carolinae, Tome 51 (2010) no. 4, pp. 595-604 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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In this note we investigate a relationship between the boundary behavior of power series and the composition of formal power series. In particular, we prove that the composition domain of a formal power series $g$ is convex and balanced which implies that the subset $\overline{\mathbb X}_g $ consisting of formal power series which can be composed by a formal power series $g$ possesses such properties. We also provide a necessary and sufficient condition for the superposition operator $T_g$ to map $\overline{\mathbb X}_g$ into itself or to map ${\mathbb X}_g$ into itself, respectively.
In this note we investigate a relationship between the boundary behavior of power series and the composition of formal power series. In particular, we prove that the composition domain of a formal power series $g$ is convex and balanced which implies that the subset $\overline{\mathbb X}_g $ consisting of formal power series which can be composed by a formal power series $g$ possesses such properties. We also provide a necessary and sufficient condition for the superposition operator $T_g$ to map $\overline{\mathbb X}_g$ into itself or to map ${\mathbb X}_g$ into itself, respectively.
Classification : 13F25, 40A30, 52A05
Keywords: composition; end behavior of convergence of power series; convex and balanced set; formal power series 
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Gan, Xiao-Xiong; Bugajewski, Dariusz. A note on formal power series. Commentationes Mathematicae Universitatis Carolinae, Tome 51 (2010) no. 4, pp. 595-604. http://geodesic.mathdoc.fr/item/CMUC_2010_51_4_a4/