A note on (weak) phase and norm retrievable Real Hilbert space frames and projections

TL;DR

This paper resolves longstanding open problems in weak phase retrieval in real Hilbert spaces, providing classifications, examples, and distinctions between phase retrieval, weak phase retrieval, and norm retrieval, especially via frames and projections.

Contribution

It offers a complete classification of vectors in r^3 for weak phase retrieval, characterizes frames doing weak phase retrieval, and introduces weak phase retrieval by projections with foundational properties.

Findings

Classified vectors in r^3 for weak phase retrieval.

Frames doing weak phase retrieval must span r^n.

Examples showing differences between phase retrieval and weak phase retrieval.

Abstract

\begin{abstract} In this manuscript, we answer a list of longstanding open problems on weak phase retrieval including: (1) A complete classification of the vectors $\{x_i\}_{i=1}^2$ in $\RR^3$ that do weak phase retrieval; (2) We show that frames doing weak phase retrieval in $\RR^n$ must span $\RR^n$; (3) We give an example of a set of vectors doing phase retrieval but their orthogonal complement hyperplanes fail weak phase retrieval; (4) We give a classification of weak phase retrievable frames - which makes clear the difference between phase retrieval and weak phase retrieval; (5) We classify when weak phase retrievable frames also do norm retrieval. We then introduce the notion of weak phase retrieval by projections and develop their basic properties. We then look at phase (norm) retrieval by projections. We end with some open problems. We provide numerous examples to show that our results are best possible. \end{abstract}