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Keywords:
composition; end behavior of convergence of power series; convex and balanced set; formal power series
composition; end behavior of convergence of power series; convex and balanced set; formal power series
Summary:
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.
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