Time: Feb 17, 2023 09:00 AM Central Time (US and Canada)

Join Zoom Meeting https://unt.zoom.us/j/86082352169

Stable phase retrieval in function spaces, Part II Mitchell A. Taylor (UC Berkeley)

Abstract: https://researchseminars.org/talk/BanachWebinars/73/

Let $(\Omega,\Sigma,\mu)$ be a measure space, and $1\leq p\leq \infty$. A subspace $E\subseteq L_p(\mu)$ is said to do stable phase retrieval (SPR) if there exists a constant $C\geq 1$ such that for any $f,g\in E$ we have $$\inf_{|\lambda|=1} |f-\lambda g|\leq C||f|-|g||.$$ In this case, if $|f|$ is known, then $f$ is uniquely determined up to an unavoidable global phase factor $\lambda$; moreover, the phase recovery map is $C$-Lipschitz. Phase retrieval appears in several applied circumstances, ranging from crystallography to quantum mechanics.

In this talk, I will present some elementary examples of subspaces of $L_p(\mu)$ which do stable phase retrieval, and discuss the structure of this class of subspaces. This is based on a joint work with M. Christ and B. Pineau, as well as a joint work with D. Freeman, B. Pineau and T. Oikhberg.