Codes and Expansions (CodEx) Seminar

Mitchell Taylor (UC Berkeley):
Stable phase retrieval in function spaces

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 \begin{equation} \inf _{|\lambda |=1} \|f-\lambda g\|\leq C\||f|-|g|\|. \end{equation} 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, T. Oikhberg and B. Pineau.