An elementary proof of a fundamental result in phase retrieval

Peter G. Casazza

Published 2020-10-14Version 1

Edidin [3] proved a fundamental result in phase retrieval: Theorem: A family of orthogonal projections $\{P_i\}_{i=1}^m$ does phase retrieval in $\mathbb{R}^n$ if and only if for every $0\not= x\in \mathbb{R}^n$, the family $\{P_ix\}_{i=1}^m$ spans $\mathbb{R}^n$. The proof of this result relies on Algebraic Geometry and so is inaccessible to many people in the field. We will give an elementary proof of this result without Algebraic Geometry. We will also solve the complex version of this result by showing that the "if" part fails and the "only if" part holds in $\mathbb{C}^n$.