Journal of Fourier Analysis and Applications

, Volume 25, Issue 6, pp 3154–3173 | Cite as

Frame Phase-Retrievability and Exact Phase-Retrievable Frames

  • Deguang HanEmail author
  • Ted Juste
  • Youfa Li
  • Wenchang Sun


A phase-retrievable frame \(\{f_{i}\}_{i}^{N}\) for an n-dimensional Hilbert space is exact if it fails to be phase-retrievable when removing any element from the frame sequence. Unlike exact frames, exact phase-retrievable frames could have different lengths. We shall prove that for the real Hilbert space case, exact phase-retrievable frame of length N exists for every \(2n-1\le N\le n(n+1)/2\). For arbitrary frames we introduce the concept of redundancy with respect to its phase-retrievability and the concept of frames with exact PR-redundancy. We investigate the phase-retrievability by studying its maximal phase-retrievable subspaces with respect to a given frame which is not necessarily phase-retrievable. These maximal PR-subspaces could have different dimensions. We are able to identify the one with the largest dimension, which can be considered as a generalization of the characterization for phase-retrievable frames. In the basis case, we prove that if M is a k-dimensional PR-subspace, then \(|supp(x)| \ge k\) for every nonzero vector \(x\in M\). Moreover, if \(1\le k< [(n+1)/2]\), then a k-dimensional PR-subspace is maximal if and only if there exists a vector \(x\in M\) such that \(|supp(x) | = k\).


Frames Phase retrieval Exact phase-retrievable frames PR-redundancy Phase-retrievable subspaces 

Mathematics Subject Classification

Primary 42C15 46C05 



The authors thank the referee very much for carefully reading the paper and for several valuable comments and suggestions. D. Han acknowledges the support from NSF under the Grant DMS-1712602. Y. Li is partially supported by the National Natural Science Foundation of China (Nos. 61561006, 11501132), the Natural Science Foundation of Guangxi (No. 2016GXNSFAA380049) and the talent project of Education Department of Guangxi Government for Young-Middle-Aged backbone teachers. W. Sun acknowledges the support from the National Natural Science Foundation of China (Nos. 11525104 and 11531013).


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Deguang Han
    • 1
    Email author
  • Ted Juste
    • 1
  • Youfa Li
    • 2
  • Wenchang Sun
    • 3
  1. 1.Department of MathematicsUniversity of Central FloridaOrlandoUSA
  2. 2.College of Mathematics and Information ScienceGuangxi UniversityNaningChina
  3. 3.School of Mathematical Sciences and LPMCNankai UniversityTianjinChina

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