Arithmetic progressions and holomorphic phase retrieval
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915270771081216 |
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| author | Liehr, Lukas |
| author_facet | Liehr, Lukas |
| contents | We study the determination of a holomorphic function from its absolute value. Given a parameter $θ\in \mathbb{R}$, we derive the following characterization of uniqueness in terms of rigidity of a set $Λ\subseteq \mathbb{R}$: if $\mathcal{F}$ is a vector space of entire functions containing all exponentials $e^{ξz}, \, ξ\in \mathbb{C} \setminus \{ 0 \}$, then every $F \in \mathcal{F}$ is uniquely determined up to a unimodular phase factor by $\{|F(z)| : z \in e^{iθ}(\mathbb{R} + iΛ)\}$ if and only if $Λ$ is not contained in an arithmetic progression $a\mathbb{Z}+b$. Leveraging this insight, we establish a series of consequences for Gabor phase retrieval and Pauli-type uniqueness problems. For instance, $\mathbb{Z} \times \tilde{\mathbb{Z}}$ is a uniqueness set for the Gabor phase retrieval problem in $L^2(\mathbb{R}_+)$, provided that $\tilde{\mathbb{Z}}$ is a suitable perturbation of the integers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_05722 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Arithmetic progressions and holomorphic phase retrieval Liehr, Lukas Complex Variables Functional Analysis 30D20, 30A99, 94A20 We study the determination of a holomorphic function from its absolute value. Given a parameter $θ\in \mathbb{R}$, we derive the following characterization of uniqueness in terms of rigidity of a set $Λ\subseteq \mathbb{R}$: if $\mathcal{F}$ is a vector space of entire functions containing all exponentials $e^{ξz}, \, ξ\in \mathbb{C} \setminus \{ 0 \}$, then every $F \in \mathcal{F}$ is uniquely determined up to a unimodular phase factor by $\{|F(z)| : z \in e^{iθ}(\mathbb{R} + iΛ)\}$ if and only if $Λ$ is not contained in an arithmetic progression $a\mathbb{Z}+b$. Leveraging this insight, we establish a series of consequences for Gabor phase retrieval and Pauli-type uniqueness problems. For instance, $\mathbb{Z} \times \tilde{\mathbb{Z}}$ is a uniqueness set for the Gabor phase retrieval problem in $L^2(\mathbb{R}_+)$, provided that $\tilde{\mathbb{Z}}$ is a suitable perturbation of the integers. |
| title | Arithmetic progressions and holomorphic phase retrieval |
| topic | Complex Variables Functional Analysis 30D20, 30A99, 94A20 |
| url | https://arxiv.org/abs/2308.05722 |