Convergence of linear combinations of iterates of an inner function

TL;DR

This paper characterizes the convergence of linear combinations of iterates of a specific class of inner functions on the unit circle, linking it to the square-summability of coefficients and properties of the resulting functions.

Contribution

It establishes necessary and sufficient conditions for convergence and boundedness of series of iterates, connecting these to classical function spaces and mean oscillation.

Findings

Series converges almost everywhere if and only if coefficients are square-summable.

Function F has bounded mean oscillation under the square-summability condition.

F is bounded on the unit disc if and only if coefficients are absolutely summable.

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 $\sum a_{n}f^{n}(\xi)$ converges at almost every point $\xi$ 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.