Completeness of systems of inner functions
Abstract
For two inner functions $\vartheta,φ\in H^\infty$, we give a simple sufficient condition for the system $\vartheta^m,\; φ^n$, $m,n\in\mathbb{Z}$, to be complete in the weak-$^*$ topology of $L^\infty(\mathbb{T})$. To be precise, we show that this system is complete whenever there is an arc $I$ of the unit circle $\mathbb{T}$ such that $\vartheta$ is univalent on $I$ and $φ$ is univalent on $\mathbb{T}\setminus I$. As an application of this result, we describe a class of analytic curves $Γ$ such that $(Γ, \mathcal{X})$ is a Heisenberg uniqueness pair, where $\mathcal{X}$ is the lattice cross $\{(m,n)\in\mathbb{Z}^2:\, mn=0\}$. Our main result extends a theorem of Hedenmalm and Montes-Rodríguez for atomic inner functions with one singularity.
Fields of Study & Subjects
Holomorphic and Operator Theory
Advanced Banach Space Theory
Geometry and complex manifolds
Publication details
- Journal / source
- Journal of Mathematical Analysis and Applications
- Volume
- 561
- Issue
- 2
- Pages
- 130683–130683
- Publication date
- 7 April 2026
- Language
- EN