Convergence of linear combinations of iterates of an inner function

Let f be an inner function with f(0)=0 which is not a rotation and let fn be its n-th iterate. Let {an} be a sequence of complex numbers. We prove that the series n-ary sumation anfn(xi) converges at almost every point xi of the unit circle if and only if n-ary sumation |an|2<infinity. The main step in the proof is to show that under this assumption, the function F= n-ary sumation anfn has bounded mean oscillation. We also prove that F is bounded on the unit disc if and only if n-ary sumation |an|<infinity. Finally we describe the sequences of coefficients {an} such that F belongs to other classical function spaces, as the disc algebra and the Dirichlet class.(c) 2022 Elsevier Masson SAS. All rights reserved.