Phase retrieval on circles and lines
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909190451101696 |
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| author | Chalendar, I. Partington, J. R. |
| author_facet | Chalendar, I. Partington, J. R. |
| contents | Let $f$ and $g$ be analytic functions on the open unit disc $\mathbb D$ such that $|f|=|g|$ on a set $A$. We give an alternative proof of the result of Perez that there exists $c$ in the unit circle $\mathbb T$ such that $f=cg$ when $A$ is the union of two lines in $\mathbb D$ intersecting at an angle that is an irrational multiple of $π$, and from this deduce a sequential generalization of the result. Similarly, the same conclusion is valid when $f$ and $g$ are in the Nevanlinna class and $A$ is the union of the unit circle and an interior circle, tangential or not. We also provide sequential versions of this result and analyse the case $A=r\mathbb T$. Finally, we examine the most general situation when there is equality on two distinct circles in the disc, providing a result or counterexample for each possible configuration. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_16255 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Phase retrieval on circles and lines Chalendar, I. Partington, J. R. Complex Variables Functional Analysis 30D05, 30H10, 94A12 Let $f$ and $g$ be analytic functions on the open unit disc $\mathbb D$ such that $|f|=|g|$ on a set $A$. We give an alternative proof of the result of Perez that there exists $c$ in the unit circle $\mathbb T$ such that $f=cg$ when $A$ is the union of two lines in $\mathbb D$ intersecting at an angle that is an irrational multiple of $π$, and from this deduce a sequential generalization of the result. Similarly, the same conclusion is valid when $f$ and $g$ are in the Nevanlinna class and $A$ is the union of the unit circle and an interior circle, tangential or not. We also provide sequential versions of this result and analyse the case $A=r\mathbb T$. Finally, we examine the most general situation when there is equality on two distinct circles in the disc, providing a result or counterexample for each possible configuration. |
| title | Phase retrieval on circles and lines |
| topic | Complex Variables Functional Analysis 30D05, 30H10, 94A12 |
| url | https://arxiv.org/abs/2403.16255 |