An elementary proof of a fundamental result in phase retrieval

2020/10/14 by Peter G. Casazza, Casazza, Peter G., Janet C. Tremain +1
Biochemistry, Genetics and Molecular Biology · Materials Science · Physics and Astronomy · #42C15 #Advanced Electron Microscopy Techniques and Applications #Advanced X-ray Imaging Techniques #Electron and X-Ray Spectroscopy Techniques #FOS: Mathematics #Functional Analysis (math.FA)

paper · pdf · doi:10.48550/arxiv.2010.07058

openalex publication_date 2020/10/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Edidin [3] proved a fundamental result in phase retrieval: Theorem: A family of orthogonal projections \P_i\_i=1^m does phase retrieval in ℝⁿ if and only if for every 0\not= x∈ ℝⁿ, the family \P_ix\_i=1^m spans ℝⁿ. 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 ℂⁿ. Finally, we will show that these techniques can be used to verify two classifications of norm retrieval.

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