### Stable phase retrieval for infinite dimensional subspaces of L_2(R)

- Series
- Analysis Seminar
- Time
- Wednesday, March 4, 2020 - 13:55 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Daniel Freeman – St. Louis University – daniel.freeman@slu.edu

The problem of phase retrieval for a set of functions $H$ can be thought of as being able to identify a function $f\in H$ or $-f\in H$ from the absolute value $|f|$. Phase retrieval for a set of functions is called stable if when $|f|$ and $|g|$ are close then $f$ is proportionally close to $g$ or $-g$. That is, we say that a set $H\subseteq L_2({\mathbb R})$ does stable phase retrieval if there exists a constant $C>0$ so that

$$\min\big(\big\|f-g\big\|_{L_2({\mathbb R})},\big\|f+g\big\|_{L_2({\mathbb R})}\big)\leq C \big\| |f|-|g| \big\|_{L_2({\mathbb R})} \qquad\textrm{ for all }f,g\in H.

$$

It is known that phase retrieval for finite dimensional spaces is always stable. On the other hand, phase retrieval for infinite dimensional spaces using a frame or a continuous frame is always unstable. We prove that there exist infinite dimensional subspaces of $L_2({\mathbb R})$ which do stable phase retrieval. This is joint work with Robert Calderbank, Ingrid Daubechies, and Nikki Freeman.