Phase retrieval on circles and lines

Chalendar, Isabelle ; Partington, Jonathan R. 

Canadian mathematical bulletin, Tome 67 (2024) no. 4, pp. 927-935

Résumé

Let f and g be analytic functions on the open unit disk ${\mathbb D}$ 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 $\pi $, 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 analyze the case $A=r{\mathbb T}$. Finally, we examine the most general situation when there is equality on two distinct circles in the disk, proving a result or counterexample for each possible configuration.

Détail

DOI : 10.4153/S0008439524000304

Mots-clés : Hardy space, phase retrieval, inner function, outer function