2021/10/13 by Peter G. Casazza, F. Akrami, Casazza, P. G. +7
Computer Science · Engineering · Physics and Astronomy · #Advanced X-ray Imaging Techniques #FOS: Mathematics #Functional Analysis (math.FA) #Image Processing Techniques and Applications #Optical measurement and interference techniques
paper · pdf · doi:10.48550/arxiv.2110.06868
openalex publication_date 2021/10/13 · openalex created_date 2021/10/25 · openalex updated_date 2026/07/28
\beginabstract In this manuscript, we answer a list of longstanding open problems on weak phase retrieval including: (1) A complete classification of the vectors \xi\i=12 in \RR3 that do weak phase retrieval; (2) We show that frames doing weak phase retrieval in \RRn must span \RRn; (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. \endabstract