Approximately Dual and Pseudo-Dual Probabilistic Frames

Clayton Shonkwiler; Dongwei Chen; Emily J. King

arxiv: 2505.13885 · v2 · pith:2FSE55OLnew · submitted 2025-05-20 · 🧮 math.FA

Approximately Dual and Pseudo-Dual Probabilistic Frames

Dongwei Chen , Emily J. King , Clayton Shonkwiler This is my paper

Pith reviewed 2026-05-22 15:01 UTC · model grok-4.3

classification 🧮 math.FA

keywords probabilistic framesdual framesapproximate dualitypseudo-dualityframe redundancyHilbert space framesatomic frames

0 comments

The pith

Every probabilistic frame has a discrete finite frame as an approximate dual.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends duality concepts from ordinary frames to probabilistic frames, which replace discrete collections of vectors with probability measures on a Hilbert space. It defines approximately dual probabilistic frames to relax exact duality while still supporting approximate reconstruction of signals, and introduces pseudo-dual frames as a further loosening of the conditions. Structural theorems establish that zero-redundancy probabilistic frames have a unique pushforward-type dual given by the canonical one, while finite-redundancy cases must be atomic and finite. The central result proves that any probabilistic frame admits a discrete finite frame as an approximate dual.

Core claim

Probabilistic frames generalize frames by using measures instead of discrete sequences. Approximately dual probabilistic frames are defined so that the analysis and synthesis operators compose to a bounded invertible operator close to the identity, preserving reconstruction up to a controlled error. The paper shows this relaxation always admits a discrete finite frame as the approximating dual, and that pseudo-duality relaxes the definition even further while retaining basic structural features.

What carries the argument

Approximately dual probabilistic frame, which replaces exact duality by requiring that the composition of the frame operator and its dual approximates the identity operator on the Hilbert space.

If this is right

Probabilistic frames with finite redundancy are necessarily atomic and finite.
For zero-redundancy probabilistic frames the canonical dual is the only dual of pushforward type.
Pseudo-dual frames extend the approximate-dual notion while keeping basic operator properties intact.
Finite discrete frames can always serve as practical approximate duals for any probabilistic frame.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Finite approximations may turn continuous probabilistic frames into objects that are easier to implement in numerical signal-processing algorithms.
The same relaxation technique could be tested on other measure-based generalizations of frames arising in integral geometry or random processes.
One could ask whether the approximation rate depends on the smoothness or support properties of the underlying probability measure.

Load-bearing premise

The new definitions of approximately dual and pseudo-dual probabilistic frames still capture the essential reconstruction and structural properties that make duality useful in the usual Hilbert-space setting of frame theory.

What would settle it

Exhibit a specific probability measure on a Hilbert space such that no sequence of discrete finite frames can make the reconstruction error arbitrarily small.

read the original abstract

This paper studies properties of dual probabilistic frames -- in particular in relation to redundancy -- and introduces both approximately dual probabilistic frames and pseudo-dual probabilistic frames. We show that the canonical dual probabilistic frame is the only dual frame of pushforward type of a probabilistic frame with zero redundancy. Furthermore, we show that probabilistic frames with finite redundancy are atomic and finite. Approximately dual probabilistic frames generalize duality, with pseudo-duality being a further generalization. We introduce these concepts and prove certain structural results. In particular, every probabilistic frame has a discrete finite frame as an approximate dual.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines approximately dual and pseudo-dual probabilistic frames and proves every probabilistic frame has a discrete finite frame as an approximate dual, plus some clean results on redundancy.

read the letter

The core contribution is the introduction of approximately dual probabilistic frames and the further generalization to pseudo-dual ones, together with the result that every probabilistic frame admits a discrete finite frame as an approximate dual. They also show that the canonical dual is the unique dual of pushforward type when redundancy is zero, and that finite redundancy forces the frame to be atomic and finite. These are concrete structural statements that extend standard frame theory without obvious contradictions in the Hilbert-space setting. The definitions appear to preserve the essential reconstruction properties while relaxing exact duality, which is a reasonable way to handle approximate cases. The proofs are not reproduced in the abstract, but the claims line up with the stress-test note that no internal inconsistency shows up from the given information. The work is incremental rather than foundational; it does not claim to solve open problems in signal processing or change how people compute frames in practice. The main limitation is that the generalizations rest on the new definitions being useful, and it is not yet clear how much extra mileage they give over existing approximate-dual notions in the literature. A reader already comfortable with probabilistic frames will see the value in the redundancy theorems and the existence result. Someone outside frame theory will find it narrow. The paper is coherent on its own terms and engages the literature through standard citations, so it meets the bar for a serious referee even if the revisions end up being mostly clarifications of the new concepts.

Referee Report

2 major / 2 minor

Summary. This paper studies dual probabilistic frames in Hilbert spaces with emphasis on redundancy, introducing the notions of approximately dual probabilistic frames and pseudo-dual probabilistic frames as successive generalizations of standard duality. It proves that the canonical dual is the unique dual of pushforward type for zero-redundancy probabilistic frames, that finite-redundancy probabilistic frames are necessarily atomic and finite, and that every probabilistic frame admits a discrete finite frame as an approximate dual.

Significance. If the structural results and existence theorem hold with rigorous proofs, the work meaningfully extends classical frame theory to the probabilistic setting by providing new duality concepts that preserve reconstruction properties while accommodating redundancy. The finiteness and atomicity results for finite-redundancy cases and the existence of discrete finite approximate duals are potentially useful for applications in signal processing or quantum information where probabilistic frames arise.

major comments (2)

The uniqueness claim for the canonical dual in the zero-redundancy pushforward case (stated in the abstract and presumably proved in the main body) is load-bearing for the subsequent generalization to approximate duality; the proof should explicitly verify that no other pushforward-type dual exists under the given measure-theoretic assumptions on the probabilistic frame.
The assertion that every probabilistic frame has a discrete finite frame as an approximate dual (central existence result) relies on the newly introduced definition of approximate duality; the manuscript must confirm that this definition reduces to ordinary duality when the approximation parameter tends to zero and that the reconstruction identity is preserved up to a controllable error term.

minor comments (2)

Notation for probabilistic frames, pushforward measures, and the new duality concepts should be introduced with a dedicated preliminary section or table to avoid ambiguity when comparing to classical frame operators.
The manuscript would benefit from an explicit statement of the Hilbert-space inner-product reconstruction formula for the newly defined approximately dual frames, including the precise error bound.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each major comment below and will make the corresponding revisions to strengthen the presentation.

read point-by-point responses

Referee: The uniqueness claim for the canonical dual in the zero-redundancy pushforward case (stated in the abstract and presumably proved in the main body) is load-bearing for the subsequent generalization to approximate duality; the proof should explicitly verify that no other pushforward-type dual exists under the given measure-theoretic assumptions on the probabilistic frame.

Authors: We agree that the uniqueness proof benefits from greater explicitness with respect to the measure-theoretic setting. In the revised version we will expand the argument in the relevant theorem to include a direct verification that any other pushforward-type measure satisfying the duality relation must coincide with the canonical dual; the argument proceeds by showing that a different pushforward would produce a strictly positive discrepancy in the frame operator integral, contradicting zero redundancy. This addition clarifies the result without changing its statement. revision: yes
Referee: The assertion that every probabilistic frame has a discrete finite frame as an approximate dual (central existence result) relies on the newly introduced definition of approximate duality; the manuscript must confirm that this definition reduces to ordinary duality when the approximation parameter tends to zero and that the reconstruction identity is preserved up to a controllable error term.

Authors: We thank the referee for this observation. We will insert a short proposition immediately after the definition of approximate duality that establishes the required limiting behavior: as the approximation parameter ε tends to zero the approximate-duality condition recovers the exact duality identity, and the reconstruction error is bounded by a constant (depending only on the frame bounds) times ε. The proof of the existence theorem will then cite this proposition to confirm consistency with the classical case. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via new definitions and standard frame theory

full rationale

The paper introduces definitions of approximately dual and pseudo-dual probabilistic frames as generalizations of standard duality, then proves structural results including uniqueness of the canonical dual for zero-redundancy pushforward cases, atomicity and finiteness for finite-redundancy frames, and existence of a discrete finite frame as an approximate dual for every probabilistic frame. These follow from the new definitions combined with properties of Hilbert-space frames and redundancy analysis; no quoted step reduces a prediction or central claim to a fitted input, self-definition, or unverified self-citation chain by construction. The central existence result is presented as a direct consequence of the introduced concepts and prior independent frame theory, making the derivation self-contained without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review is based solely on the abstract; no explicit free parameters, new entities, or ad-hoc axioms are mentioned. The work relies on background assumptions from probabilistic frame theory in Hilbert spaces.

axioms (1)

domain assumption Standard definitions and properties of frames, dual frames, and probabilistic frames in Hilbert spaces hold.
The paper extends existing frame theory without re-deriving its foundations.

pith-pipeline@v0.9.0 · 5614 in / 1182 out tokens · 66191 ms · 2026-05-22T15:01:05.678704+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 4.1 and Theorem 5.6: approximately dual probabilistic frames via couplings γ with ||∫xyᵗ dγ - Id|| < 1; every probabilistic frame has a discrete finite approximate dual.
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 3.6: finite redundancy implies finitely supported atomic measure.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

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A combinatorial characterization of tight fusion frames.Pacific Journal of Mathematics, 275(2):257–294, 2015

Marcin Bownik, Kurt Luoto, and Edward Richmond. A combinatorial characterization of tight fusion frames.Pacific Journal of Mathematics, 275(2):257–294, 2015

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Martin Ehler and Kasso A. Okoudjou. Probabilistic frames: An overview. In Peter G. Casazza and Gitta Kutyniok, editors,Finite Frames: Theory and Applications, Applied and Numer- ical Harmonic Analysis, pages 415–436. Birkhäuser, Boston, MA, USA, 2013. 24 DONGWEI CHEN, EMILY J. KING, AND CLAYTON SHONKWILER

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Yonina C. Eldar. Sampling with arbitrary sampling and reconstruction spaces and oblique dual frame vectors.Journal of Fourier Analysis and Applications, 9:77–96, 2003

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Massar, and Dustin G

Matthew Fickus, Melody L. Massar, and Dustin G. Mixon. Finite frames and filter banks. In Peter G. Casazza and Gitta Kutyniok, editors,Finite Frames: Theory and Applications, Applied and Numerical Harmonic Analysis, pages 337–379. Birkhäuser, Boston, MA, USA, 2013

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Number 22 in EMS Textbooks in Mathematics

Alessio Figalli and Federico Glaudo.An Invitation to Optimal Transport, Wasserstein Dis- tances, and Gradient Flows. Number 22 in EMS Textbooks in Mathematics. EMS Press, Berlin, Germany, 2021

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Hosseini Giv and M

H. Hosseini Giv and M. Radjabalipour. On the structure and properties of lower bounded analytic frames.Iran. J. Sci. Technol. Trans. A Sci., 37(3):227–230, 2013

work page 2013
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Applied and Numerical Har- monic Analysis

Karlheinz Gröchenig.Foundations of Time-Frequency Analysis. Applied and Numerical Har- monic Analysis. Birkhäuser, Boston, MA, USA, 2001

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Applied and Numerical Har- monic Analysis

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Density and duality theorems for regular Gabor frames.J

Mads Sielemann Jakobsen and Jakob Lemvig. Density and duality theorems for regular Gabor frames.J. Funct. Anal., 270(1):229–263, 2016

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Applied and Numerical Harmonic Analysis

Gitta Kutyniok and Demetrio Labate, editors.Shearlets: Multiscale Analysis for Multivariate Data. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston, MA, USA, 2012

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Probabilistic tight frames and representa- tion of positive operator-valued measures.Applied and Computational Harmonic Analysis, 47(1):212–225, 2019

Mostafa Maslouhi and Soulaymane Loukili. Probabilistic tight frames and representa- tion of positive operator-valued measures.Applied and Computational Harmonic Analysis, 47(1):212–225, 2019

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Ramu Naidu and Chandra R

R. Ramu Naidu and Chandra R. Murthy. Construction of unimodular tight frames for com- pressed sensing using majorization-minimization.Signal Processing, 172:107516, 2020

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[29]

On the characterization of generalized dual frames.University Polytehnica of Bucharest Scientific Bulletin, Series A, 82(1):161–170, 2020

Atefe Razghandi and Ali Akbar Arefijamaal. On the characterization of generalized dual frames.University Polytehnica of Bucharest Scientific Bulletin, Series A, 82(1):161–170, 2020

work page 2020
[30]

Frames, their relatives and reproducing kernel Hilbert spaces.Journal of Physics A: Mathematical and Theoretical, 53(1):015204, 2019

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Heath Jr

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[32]

Duality and geodesics for probabilistic frames.Linear Algebra and Its Applications, 532:198–221, 2017

Clare Wickman and Kasso Okoudjou. Duality and geodesics for probabilistic frames.Linear Algebra and Its Applications, 532:198–221, 2017

work page 2017
[33]

PhD thesis, University of Maryland, College Park, 2014

Clare Georgianna Wickman.An Optimal Transport Approach to Some Problems in Frame Theory. PhD thesis, University of Maryland, College Park, 2014. Department of Mathematics, Colorado State University, Fort Collins, CO, US, 80523 Email address:dongwei.chen@colostate.edu Department of Mathematics, Colorado State University, Fort Collins, CO, US, 80523 Email a...

work page 2014

[1] [1]

Askari-Hemmat, M

A. Askari-Hemmat, M. A. Dehghan, and M. Radjabalipour. Generalized frames and their redundancy.Proc. Amer. Math. Soc., 129(4):1143–1147, 2001

work page 2001

[2] [2]

On excesses of frames.Glasnik Matematički, 50(2):415–427, 2015

Damir Bakić and Tomislav Berić. On excesses of frames.Glasnik Matematički, 50(2):415–427, 2015

work page 2015

[3] [3]

Casazza, Christopher Heil, and Zeph Landau

Radu Balan, Peter G. Casazza, Christopher Heil, and Zeph Landau. Deficits and excesses of frames.Advances in Computational Mathematics, 18:93–116, 2003

work page 2003

[4] [4]

The algebraic ma- troid of the finite unit norm tight frame (funtf) variety.Journal of Pure and Applied Algebra, 224(8):106351, 2020

Daniel Irving Bernstein, Cameron Farnsworth, and Jose Israel Rodriguez. The algebraic ma- troid of the finite unit norm tight frame (funtf) variety.Journal of Pure and Applied Algebra, 224(8):106351, 2020

work page 2020

[5] [5]

A combinatorial characterization of tight fusion frames.Pacific Journal of Mathematics, 275(2):257–294, 2015

Marcin Bownik, Kurt Luoto, and Edward Richmond. A combinatorial characterization of tight fusion frames.Pacific Journal of Mathematics, 275(2):257–294, 2015

work page 2015

[6] [6]

Algebraic geometry and finite frames

Jameson Cahill and Nate Strawn. Algebraic geometry and finite frames. In Peter G. Casazza and Gitta Kutyniok, editors,Finite Frames: Theory and Applications, Applied and Numer- ical Harmonic Analysis, pages 141–170. Birkhäuser, Boston, MA, USA, 2013

work page 2013

[7] [7]

Peter G. Casazza. The Kadison–Singer and Paulsen problems in finite frame theory. In Pe- terG.CasazzaandGittaKutyniok, editors,Finite Frames: Theory and Applications, Applied and Numerical Harmonic Analysis, pages 381–413. Birkhäuser, Boston, MA, USA, 2013

work page 2013

[8] [8]

Paley–Wiener theorem for probabilistic frames

Dongwei Chen. Paley–Wiener theorem for probabilistic frames. Preprint, 2023, arXiv:2310.17830 [math.FA]

work page arXiv 2023

[9] [9]

Probabilistic frames and Wasserstein distances

Dongwei Chen and Martin Schmoll. Probabilistic frames and Wasserstein distances. Preprint, 2025,arXiv:2501.02602 [math.PR]

work page arXiv 2025

[10] [10]

Applied and Numerical Har- monic Analysis

Ole Christensen.An Introduction to Frames and Riesz Bases. Applied and Numerical Har- monic Analysis. Springer, Cham, 2016

work page 2016

[11] [11]

Operator representations of frames: Boundedness, duality, and stability.Integral Equations and Operator Theory, 88:483–499, 2017

Ole Christensen and Marzieh Hasannasab. Operator representations of frames: Boundedness, duality, and stability.Integral Equations and Operator Theory, 88:483–499, 2017

work page 2017

[12] [12]

Laugesen

Ole Christensen and Richard S. Laugesen. Approximately dual frames in Hilbert spaces and applications to Gabor frames.Sampling Theory in Signal and Image Processing, 9(1–3):77– 90, 2010

work page 2010

[13] [13]

Elsevier, 2000

Kai Lai Chung.A course in probability theory. Elsevier, 2000

work page 2000

[14] [14]

IngridDaubechies.Ten Lectures on Wavelets.Number61inCBMS–NSFRegionalConference SeriesinAppliedMathematics.SocietyforIndustrialandAppliedMathematics, Philadelphia, PA, USA, 1992

work page 1992

[15] [15]

RichardJ.DuffinandAlbertC.Schaeffer.AclassofnonharmonicFourierseries.Transactions of the American Mathematical Society, 72(2):341–366, 1952

work page 1952

[16] [16]

Random tight frames.Journal of Fourier Analysis and Applications, 18(1):1– 20, 2012

Martin Ehler. Random tight frames.Journal of Fourier Analysis and Applications, 18(1):1– 20, 2012

work page 2012

[17] [17]

Okoudjou

Martin Ehler and Kasso A. Okoudjou. Probabilistic frames: An overview. In Peter G. Casazza and Gitta Kutyniok, editors,Finite Frames: Theory and Applications, Applied and Numer- ical Harmonic Analysis, pages 415–436. Birkhäuser, Boston, MA, USA, 2013. 24 DONGWEI CHEN, EMILY J. KING, AND CLAYTON SHONKWILER

work page 2013

[18] [18]

Yonina C. Eldar. Sampling with arbitrary sampling and reconstruction spaces and oblique dual frame vectors.Journal of Fourier Analysis and Applications, 9:77–96, 2003

work page 2003

[19] [19]

Massar, and Dustin G

Matthew Fickus, Melody L. Massar, and Dustin G. Mixon. Finite frames and filter banks. In Peter G. Casazza and Gitta Kutyniok, editors,Finite Frames: Theory and Applications, Applied and Numerical Harmonic Analysis, pages 337–379. Birkhäuser, Boston, MA, USA, 2013

work page 2013

[20] [20]

Number 22 in EMS Textbooks in Mathematics

Alessio Figalli and Federico Glaudo.An Invitation to Optimal Transport, Wasserstein Dis- tances, and Gradient Flows. Number 22 in EMS Textbooks in Mathematics. EMS Press, Berlin, Germany, 2021

work page 2021

[21] [21]

Hosseini Giv and M

H. Hosseini Giv and M. Radjabalipour. On the structure and properties of lower bounded analytic frames.Iran. J. Sci. Technol. Trans. A Sci., 37(3):227–230, 2013

work page 2013

[22] [22]

Applied and Numerical Har- monic Analysis

Karlheinz Gröchenig.Foundations of Time-Frequency Analysis. Applied and Numerical Har- monic Analysis. Birkhäuser, Boston, MA, USA, 2001

work page 2001

[23] [23]

Applied and Numerical Har- monic Analysis

Christopher Heil.A Basis Theory Primer: Expanded Edition. Applied and Numerical Har- monic Analysis. Birkhäuser, Boston, MA, USA, 2011

work page 2011

[24] [24]

Density and duality theorems for regular Gabor frames.J

Mads Sielemann Jakobsen and Jakob Lemvig. Density and duality theorems for regular Gabor frames.J. Funct. Anal., 270(1):229–263, 2016

work page 2016

[25] [25]

Some properties of approximately dual frames in Hilbert spaces.Results in Mathematics, 70:475–485, 2016

Hossein Javanshiri. Some properties of approximately dual frames in Hilbert spaces.Results in Mathematics, 70:475–485, 2016

work page 2016

[26] [26]

Applied and Numerical Harmonic Analysis

Gitta Kutyniok and Demetrio Labate, editors.Shearlets: Multiscale Analysis for Multivariate Data. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston, MA, USA, 2012

work page 2012

[27] [27]

Probabilistic tight frames and representa- tion of positive operator-valued measures.Applied and Computational Harmonic Analysis, 47(1):212–225, 2019

Mostafa Maslouhi and Soulaymane Loukili. Probabilistic tight frames and representa- tion of positive operator-valued measures.Applied and Computational Harmonic Analysis, 47(1):212–225, 2019

work page 2019

[28] [28]

Ramu Naidu and Chandra R

R. Ramu Naidu and Chandra R. Murthy. Construction of unimodular tight frames for com- pressed sensing using majorization-minimization.Signal Processing, 172:107516, 2020

work page 2020

[29] [29]

On the characterization of generalized dual frames.University Polytehnica of Bucharest Scientific Bulletin, Series A, 82(1):161–170, 2020

Atefe Razghandi and Ali Akbar Arefijamaal. On the characterization of generalized dual frames.University Polytehnica of Bucharest Scientific Bulletin, Series A, 82(1):161–170, 2020

work page 2020

[30] [30]

Frames, their relatives and reproducing kernel Hilbert spaces.Journal of Physics A: Mathematical and Theoretical, 53(1):015204, 2019

Michael Speckbacher and Peter Balazs. Frames, their relatives and reproducing kernel Hilbert spaces.Journal of Physics A: Mathematical and Theoretical, 53(1):015204, 2019

work page 2019

[31] [31]

Heath Jr

Thomas Strohmer and Robert W. Heath Jr. Grassmannian frames with applications to coding and communication.Applied and Computational Harmonic Analysis, 14(3):257–275, 2003

work page 2003

[32] [32]

Duality and geodesics for probabilistic frames.Linear Algebra and Its Applications, 532:198–221, 2017

Clare Wickman and Kasso Okoudjou. Duality and geodesics for probabilistic frames.Linear Algebra and Its Applications, 532:198–221, 2017

work page 2017

[33] [33]

PhD thesis, University of Maryland, College Park, 2014

Clare Georgianna Wickman.An Optimal Transport Approach to Some Problems in Frame Theory. PhD thesis, University of Maryland, College Park, 2014. Department of Mathematics, Colorado State University, Fort Collins, CO, US, 80523 Email address:dongwei.chen@colostate.edu Department of Mathematics, Colorado State University, Fort Collins, CO, US, 80523 Email a...

work page 2014