Convergence of linear combinations of iterates of an inner function

Artur Nicolau

paper

Convergence of linear combinations of iterates of an inner function

arXiv:2103.07238

Abstract

Let $f$ be an inner function with $f (0) = 0$ which is not a rotation and let $f^{n}$ be its $n$ -th iterate. Let ${a_{n}}$ be a sequence of complex numbers. We prove that the series $\sum a_{n} f^{n} (ξ)$ converges at almost every point $ξ$ of the unit circle if and only if $\sum ∣ a_{n} ∣^{2} < \infty$ . The main step in the proof is to show that under this assumption, the function $F = \sum a_{n} f^{n}$ has bounded mean oscillation. We also prove that $F$ is bounded on the unit disc if and only if $\sum ∣ a_{n} ∣ < \infty$ . Finally we describe the sequences of coefficients ${a_{n}}$ such that $F$ belongs to other classical function spaces, as the disc algebra and the Dirichlet class.