Norm Retrieval and Phase Retrieval by Projections

Casazza, Peter G.; Ghoreishi, Dorsa; Jose, Shani; Tremain, Janet C.

doi:10.3390/axioms6010006

Cited by 19 publications

(9 citation statements)

References 19 publications

(29 reference statements)

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“…For the case of finite-dimensional, various aspects to this problem which include the uniqueness and the stability of solutions were well studied [4-7, 10, 12, 15, 17-20, 24, 25, 31]. Further generalizations including norm retrieval [3,11] and phase retrieval from projections [9,15] were also studied.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Local and Global Phaseless Sampling in Real Spline Spaces

Sun

2017

Preprint

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We study the recovery of functions in real spline spaces from unsigned sampled values. We consider two types of recovery. The one is to recover functions locally from finitely many unsigned samples. And the other is to recover functions on the whole line from infinitely many unsigned samples. In both cases, we give characterizations for a sequence of distinct points to be a phaseless sampling sequence, at which any nonseparable function is determined up to a sign on an interval or on the whole line by its unsigned sampled values. Moreover, for the case of local recovery, we also study the almost phaseless sampling and give a necessary and sufficient condition for a sequence of points to admit local recovery for almost all functions.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Local and Global Phaseless Sampling in Real Spline Spaces

Sun

2017

Preprint

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“…It was shown that with certain frames, we can recover a signal up to a phase from its phaseless frame coefficients. We refer to [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] for various aspects on this topic.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Phaseless Sampling and Linear Reconstruction of Functions in Spline Spaces

Sun¹

2017

Preprint

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We study phaseless sampling in spline spaces generated by Bsplines with arbitrary knots. For real spline spaces, we give a necessary and sufficient condition for a sequence of sampling points to admit a local phase retrieval of any nonseparable function. We also study phaseless sampling in complex spline spaces and illustrate that phase retrieval is impossible in this case. Nevertheless, we show that phaseless sampling is possible. For any function f in a complex spline space, no mater it is separable or not, we show that |f (x)| 2 is uniquely determined and can be recovered linearly from its sampled values at a well chosen sequence of sampling points. We give necessary and sufficient conditions for such sequences.generated by the m-degree B-spline ϕ m = χ [0,1] * . . . * χ [0,1] ( m + 1 terms), m ≥ 1,

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“…(2) For every 0 = x ∈ R n , x ∈ span{P i x} m i=1 . The following example appears in [9]. We will give a new proof which generalizes to answer another problem.…”

Section: Hyperplanesmentioning

confidence: 98%

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

Akrami¹,

Casazza²,

Fard³

et al. 2021

Preprint

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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 } m i=1 in R n are not dense in the family of m ≤ (2n − 2)-element sets of vectors in R n for every finite n and the families of vectors which do norm retrieval in ℓ 2 are not dense in the infinite families of vectors in ℓ 2 . We also show that if a Riesz basis does norm retrieval in ℓ 2 , then it is an orthogonal sequence. We provide numerous examples to show that our results are best possible.

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Norm Retrieval and Phase Retrieval by Projections

Cited by 19 publications

References 19 publications

Local and Global Phaseless Sampling in Real Spline Spaces

Local and Global Phaseless Sampling in Real Spline Spaces

Phaseless Sampling and Linear Reconstruction of Functions in Spline Spaces

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

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