A note on phase (norm) retrievable Real Hilbert space (fusion) frames

2021/07/20 by F. Akrami, Akrami, F., Peter G. Casazza +5
Computer Science · Earth and Planetary Sciences · Engineering · #42C15 #42C40 #FOS: Mathematics #Functional Analysis (math.FA) #Image Processing Techniques and Applications #Optical measurement and interference techniques #Seismic Imaging and Inversion Techniques

paper · pdf · doi:10.48550/arxiv.2107.09606

openalex publication_date 2021/07/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this manuscript, we present several new results in finite and countable dimensional real Hilbert space phase retrieval and norm retrieval by vectors and projections. We make a detailed study of when hyperplanes do norm retrieval. Also, we show that the families of norm retrievable frames \f_i\_i=1^m in ℝⁿ are not dense in the family of m≤ (2n-2)-element sets of vectors in ℝⁿ for every finite n and the families of vectors which do norm retrieval in ℓ² are not dense in the infinite families of vectors in ℓ². We also show that if a Riesz basis does norm retrieval in ℓ², then it is an orthogonal sequence. We provide numerous examples to show that our results are best possible.

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