On a theorem of Seidel and Walsh

Abstract

Given a sequence (α _n ) ^∞ _n in$\mathbb{D}$ with$\mathop {\lim }\limits_{n \to \infty } |\alpha _n | = 1$ there are functions$f \in H(\mathbb{D})$ such that$\{ f \circ S_{\alpha _n } :n \in \mathbb{N}\} ,S_{\alpha _n } (z) = (z - \alpha _n )/(1 - \tilde \alpha _n z)$, is a dense subset of$H(\mathbb{D})$, and the set of functions with this property is residual in$H(\mathbb{D})$. We will show that in$A(\mathbb{D})$ and some related Banach spaceX there are functionsf with$\{ f' \circ S_{\alpha _n } :n \in \mathbb{N}\}$ is dense in$H(\mathbb{D})$, and we will give a sufficient condition when the set of such functions is residual inX.