Convergence of linear combinations of iterates of an inner function

2021/03/12 by Artur Nicolau, Nicolau, Artur
Mathematics · #30H10 #30J05 (Primary) #60F05 (Secondary) #Analytic and geometric function theory #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2103.07238

openalex publication_date 2021/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Let f be an inner function with f(0)=0 which is not a rotation and let fⁿ be its n-th iterate. Let \a_n\ be a sequence of complex numbers. We prove that the series ∑ a_nfⁿ(ξ) converges at almost every point ξ of the unit circle if and only if ∑ |a_n|² < ∞. The main step in the proof is to show that under this assumption, the function F= ∑ a_n fⁿ has bounded mean oscillation. We also prove that F is bounded on the unit disc if and only if ∑ |a_n| < ∞. 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.

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