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 ∑anfn(ξ) converges at almost every point ξ of the unit circle if and only if ∑|an|2<∞. The main step in the proof is to show that under t...
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| Format: | Article |
| Publication date: | 2022 |
| Country: | España |
| Institution: | Universitat Autònoma de Barcelona |
| Repository: | Dipòsit Digital de Documents de la UAB |
| Language: | English |
| OAI Identifier: | oai:ddd.uab.cat:306144 |
| Online Access: | https://ddd.uab.cat/record/306144 https://dx.doi.org/urn:doi:10.1016/j.matpur.2022.03.003 |
| Access Level: | Open access |
| Keyword: | Inner function Bounded mean oscillation Hardy spaces Dirichlet class |
| Summary: | 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 ∑anfn(ξ) converges at almost every point ξ of the unit circle if and only if ∑|an|2<∞. The main step in the proof is to show that under this assumption, the function F=∑anfn has bounded mean oscillation. We also prove that F is bounded on the unit disc if and only if ∑|an|<∞. 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. |
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