A note on norm retrievable real Hilbert space frames

Akrami, F.; Casazza, Peter G.; Fard, M. A. Hasankhani; Rahimi, Asghar

doi:10.1016/j.jmaa.2022.126620

Cited by 6 publications

(6 citation statements)

References 9 publications

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“…Proof. In case Φ does weak phase retrieval, it is simple to see that P Φ does weak phase retrieval, for every orthogonal projection P on H n , as well, see also [1]. Conversely, let x, y ∈ H n and | x, φ i | = | y, φ i |.…”

Section: Weak Phase Retrievalmentioning

confidence: 96%

“…Also, clearly phase retrieval implies weak phase retrieval property, although the converse does not hold in general. As a simple example {(1, 1), (1, −1)} does weak phase retrieval for R 2 , see [1], but clearly does not phase retrieval. Now, we state the concept of a generic set; A subset Ω ⊆ R n is called generic whenever there exists a nonzero polynomial p(x 1 , ..., x n ) so that…”

Section: Theorem 12 [8]mentioning

confidence: 99%

“…Considering D Φ as a metric space by d(G, H) = i∈I2n−1 g i − h i , we define the mapping ξ : 1) is the unique sequence associated to G ∈ D Φ as in (2.2). Then, clearly ξ is well-defined and injective.…”

Section: Phase Retrieval Dual Framesmentioning

confidence: 99%

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Characterization of (weak) phase retrieval dual frames

Arabyani-Neyshaburi¹,

Arefijamaal²,

Gol³

2023

Preprint

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Recovering a signal up to a unimodular constant from the magnitudes of linear measurements has been popular and well studied in recent years. However, numerous unsolved problems regarding phase retrieval still exist. Given a phase retrieval frame, may the family of phase retrieval dual frames be classified? And is such a family dense in the set of dual frames? Can we present the equivalent conditions for a family of vectors to do weak phase retrieval in complex Hilbert space case? What is the connection between phase, weak phase and norm retrieval? In this context, we aim to deal with these open problems concerning phase retrieval dual frames, weak phase retrieval frames, and specially investigate equivalent conditions for identifying these features. We provide some characterizations of alternate dual frames of a phase retrieval frame which yield phase retrieval in finite dimensional Hilbert spaces. Moreover, for some classes of frames, we show that the family of phase retrieval dual frames is open and dense in the set of dual frames. Then, we study weak phase retrieval problem. Among other things, we obtain some equivalent conditions on a family of vectors to do phase retrieval in terms of weak phase retrieval.

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Section: Weak Phase Retrievalmentioning

confidence: 96%

Section: Theorem 12 [8]mentioning

confidence: 99%

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Characterization of (weak) phase retrieval dual frames

Arabyani-Neyshaburi¹,

Arefijamaal²,

Gol³

2023

Preprint

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show abstract

“…Phase retrieval was introduced in reference [4]. See reference [1] for an introduction to norm retrieval.…”

Section: Preliminariesmentioning

confidence: 99%

“…This always works if 0 < inf i∈I a i ≤ sup i∈I a i < ∞. But this is not necessary in general [1]. The complement property is an essential issue here.…”

Section: Preliminariesmentioning

confidence: 99%

Classifying weak phase retrieval

Casazza¹,

Akrami²

2023

Preprint

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We will give several surprising equivalences and consequences of weak phase retrieval. These results give a complete understanding of the difference between weak phase retrieval and phase retrieval. We also answer two longstanding open problems on weak phase retrieval: (1) We show that the families of weak phase retrievable frames {x i } m i=1 in R n are not dense in the family of m-element sets of vectors in R n for all m ≥ 2n − 2; (2) We show that any frame {x i } 2n−2 i=1 containing one or more canonical basis vectors in R n cannot do weak phase retrieval. We provide numerous examples to show that the obtained results are best possible.

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Frame structure derived from a non-Blaschke sequence on the unit disc

Sreedharan,

Asharaf

2024

J Anal

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A note on norm retrievable real Hilbert space frames

Cited by 6 publications

References 9 publications

Characterization of (weak) phase retrieval dual frames

Characterization of (weak) phase retrieval dual frames

Classifying weak phase retrieval

Frame structure derived from a non-Blaschke sequence on the unit disc

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