Completeness of systems of inner functions
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914878942347264 |
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| author | Miheisi, Nazar |
| author_facet | Miheisi, Nazar |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_03076 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Completeness of systems of inner functions Miheisi, Nazar Functional Analysis Complex Variables 30B60 (Primary), 30J05 (Secondary) 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. |
| title | Completeness of systems of inner functions |
| topic | Functional Analysis Complex Variables 30B60 (Primary), 30J05 (Secondary) |
| url | https://arxiv.org/abs/2404.03076 |