Discrete Quaternionic (Multi-window) Gabor Systems

Najib Khachiaa

REVIEW 4 major objections 4 minor 18 references

Discrete Quaternionic (Multi-window) Gabor Systems

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For real-valued windows, a discrete quaternionic Gabor system is a frame exactly when a finite list of pointwise matrix inequalities holds.

desk verdict A useful real-window multi-window quaternionic Gabor frame characterization, but the paper ships lemmas with dropped conjugates and a duality theorem stated too broadly; Theorem 1 survives, the rest needs repair. read the letter →

arxiv 2411.16988 v1 pith:KC2JQV3M submitted 2024-11-25 math.FA

Najib Khachiaa This is my paper · ORCID

classification math.FA MSC 42C1542C4051F30

keywords quaternionicGaborframesmulti-windowsystemsHilbertspacesframecharacterizationParsevalorthonormalbasesdualityofstability

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves that, when the windows are real-valued, the frame property of a discrete quaternionic multi-window Gabor system on ℓ²(ℤ², ℍ) is completely determined by finite matrix data. For a system G(g,L,M,N) with windows g_l, the frame bounds A ≤ B are equivalent to the pointwise operator inequalities A/M² I ≤ Σ_l M_{g_l}(k) M_{g_l}^t(k) ≤ B/M² I holding for every k in the N-periodic residue classes. The same matrix machinery characterizes Parseval frames, orthonormal bases, duality, and stability. This matters because it converts an infinite-dimensional frame question in a noncommutative setting into a check on finitely many bi-infinite matrices.

What carries the argument

The matrix-valued function M_h : ℤ² → M(ℍ) associated with a window h, whose (p,n)-entry is h(k + pM − nN). This function converts translation-modulation sums into products of bi-infinite matrices; for real-valued windows the products M_g(k)M_g^t(k) are N-periodic in k, so the entire frame analysis reduces to checking operator inequalities on the N² residue classes modulo N.

What would settle it

Take the pure quaternionic window g = iδ_0 with M=N=1: G(δ_0,1,1,1) is an orthonormal basis with frame bounds 1, yet M_g(0)M_g^t(0) = −I, so the inequality of Theorem 1 fails, showing the real-valued hypothesis is essential. For the real-valued theorem itself, compute numerically the extremal eigenvalues of Σ_l M_{g_l}(k)M_{g_l}^t(k) over k ∈ N_N² for a randomly chosen real window and compare them with the frame bounds of G(g,L,M,N) obtained by a direct finite-lattice computation; any mismatch would refute the equivalence.

Watch

Extended reading notes

Core claim

The central discovery is that real-valued windows commute with the quaternionic modulation factors, which collapses the frame operator into a clean two-part decomposition and lets the frame bounds be read off from the spectra of the matrices S(k) := Σ_l M_{g_l}(k) M_{g_l}^t(k). Theorem 1 states that G(g,L,M,N) is a frame with bounds A ≤ B if and only if A/M² I ≤ S(k) ≤ B/M² I for all k ∈ ℤ² (equivalently, k ∈ N_N²). Theorem 3 sharpens this for Parseval frames: S(k) must equal (1/M²)I on the diagonal and vanish off it. Within the Parseval class, Theorem 4 shows the system is an orthonormal basis precisely when N² = L M², which forces L to be a perfect square; Theorem 6 proves such bases exist exactly under that parameter condition and constructs them from indicator windows. Theorem 7 gives the dual-frame analogue S_{g,h}(k) = (1/M²)I, and Theorem 8 shows that sufficiently small perturbations of the window matrices preserve the frame property with quantitative bounds.

Load-bearing premise

The equivalence rests on the windows being real-valued so they commute with the quaternionic modulation factors, and on the implicit assumption that the bi-infinite matrices M_g(k) and their sums are bounded operators on ℓ²(ℤ²).

Editorial extensions

If this is right

Frame verification for real-window quaternionic Gabor systems becomes a finite computation: check the operator inequalities on the N² residue classes rather than analyzing the whole system.
The Parseval characterization gives an explicit construction of Parseval frames from indicator windows whenever the parameter condition N² ≤ L M² holds.
Orthonormal Gabor bases exist exactly when N² = L M², and the number of windows L must be a perfect square, so the parameter pairs (L,M,N) that admit such bases are completely classified.
Duality between two real-window Gabor systems is characterized by the same matrix condition with cross products, extending the classical discrete-periodic duality theory to the quaternionic setting.
The stability theorem supplies quantitative frame bounds for perturbed systems, showing that small changes in the window matrices cannot destroy the frame property.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The real-valued hypothesis is likely removable only if one tracks noncommutativity more carefully; a two-sided modulation construction might allow complex or quaternionic windows to satisfy an analogous but more complicated matrix inequality.
The reduction to N-periodic matrix inequalities suggests a direct algorithmic route: construct quaternionic Gabor frames by solving finite-dimensional matrix inequalities on the residue classes, which could be useful in quaternionic signal processing.
The parameter condition N² = L M² is a density-type statement that mirrors the classical density theorem for Gabor systems, hinting that a full quaternionic density theory may be within reach.
The same matrix-based approach could extend to other noncommutative coefficient algebras, such as Clifford algebras, whenever the window algebra commutes with the modulation factors.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

What you should know: this is a proof-based extension of multi-window Gabor frame theory to quaternionic-valued sequences on Z^2, with the main positive result restricted to real-valued windows. Theorem 1, the centerpiece, says that for real windows the Gabor system is a frame with bounds A and B exactly when the pointwise, N-periodic matrix inequality A/M^2 I <= sum_l M_g_l(k) M_g_l^t(k) <= B/M^2 I holds. That is a genuine result, and the proof via periodizing over the modulation lattice is standard and plausible. The multi-window real-valued quaternionic combination appears to be new; the explicit Parseval-frame and orthonormal-basis constructions in Examples 1 and 2 are concrete and checkable. The density condition N^2 <= L M^2 and the perfect-square window count for ONBs also match the expected discrete Gabor pattern.

Now the soft spots, in proportion. Lemma 2 and Lemma 6 drop quaternionic conjugates in the cross terms: the off-diagonal term should be h(k) overline{h(k+pM)}, not h(k)h(k+pM), and similarly in the duality identity. For H-valued test functions the written identities are false. The good news is that Theorem 1 does not use Lemma 2, so its proof survives. The bad news is that Theorem 2 and Theorem 3 rely on Lemma 2, and Theorem 7 is stated for general quaternionic windows while its proof invokes Lemma 6, which explicitly requires real-valued windows. The Parseval-frame and duality characterizations therefore need either a repaired conjugation argument or an explicit real-window hypothesis. This is a load-bearing gap, but it looks repairable. Separately, the bi-infinite matrices in Theorem 1 are treated as bounded operators without a stated condition; arbitrary l^2 windows need not give bounded matrix multiplication. The body never cites or compares with [17] and [18], which appear to cover single-window quaternionic Gabor frames, so the novelty boundary is not clearly drawn. Typos, malformed symbols, and forward references (Theorem 1 uses Lemma 5 before it is stated) are numerous but secondary.

Verdict: this paper deserves a serious referee because the central characterization is likely correct and useful to people working on quaternionic frame theory and quaternionic signal processing. I would not cite it in its current form, but I would read a corrected version. Send it out, expect major revision, and ask the author to fix the conjugation, add the missing hypotheses, and compare explicitly with [17] and [18].

Referee Report

4 major / 4 minor

Summary. The paper studies discrete multi-window Gabor systems in the quaternionic Hilbert space ℓ²(Z×Z,H), with modulation by left-multiplication by e^{2πi m₁/M} and right-multiplication by e^{2πj m₂/M}. The central claim is Theorem 1: for real-valued windows g_l, the system is a frame with bounds A≤B if and only if the matrix-valued function ∑_l M_{g_l}(k)M_{g_l}^t(k) satisfies the pointwise operator inequality A/M² I ≼ … ≼ B/M² I on an N-periodic set of k. Subsequent sections characterize Parseval frames, orthonormal bases, dual frames, and stability under perturbations, again in terms of the matrix-valued functions M_g.

Significance. If the main theorem is correct, it gives a transparent, checkable characterization of quaternionic Gabor frames for real windows, extending a body of complex Gabor theory to a noncommutative setting where the usual commutativity arguments fail. The paper is self-contained in its reductions, uses no fitted parameters, and provides constructive examples for Parseval frames and orthonormal bases. However, the significance is currently limited by several local but load-bearing gaps: an incorrect identity in Lemma 2, a missing real-window hypothesis in Theorem 7, and an unstated boundedness condition for the infinite matrices in Theorem 1. These are repairable, but the results are not established as written.

major comments (4)

[Section 2, Lemma 2 and F2(h)] Lemma 2 is false as stated for H-valued h. In the expansion of |∑_k ⟨E^m_M T_{nN} g_l, h⟩|², the off-diagonal term must contain a quaternionic conjugate, e.g. ̅h(k)h(k+pM) (with the paper's inner-product convention), not h(k)h(k+pM). The real-valuedness of g_l makes the window commute with the exponential factors, but it does not remove the conjugate from h. This invalidates the stated identity in Lemma 2 and consequently the proofs that rely on it: Theorem 2, Theorem 3, Proposition 4, and, via Lemma 6, Theorem 7. Most of the arguments can be repaired by inserting the missing conjugate — for the Bessel bound in Theorem 2 only the modulus is used, and in Theorem 3 the two-point test still gives 1 — but the manuscript must present correct identities before these results can be accepted. I note that Theorem 1 itself proceeds through Lemma 5 rather than Lemma 2, so its proof is not invalidated by this particular error.
[Section 4, Theorem 7 and Lemma 6] Theorem 7 is stated for arbitrary g_l,h_l ∈ ℓ²(Z×Z,H), but its proof invokes Lemma 6, which is proved only under the hypothesis g_l,h_l ∈ ℓ²(Z×Z,R). The real-valued hypothesis is essential for the commutativity step that collapses the double modulation sum into M_{g_l}(k)M_{h_l}^t(k). The theorem must either be restricted to real-valued windows or be given a genuinely new proof for H-valued windows; as written, the duality characterization for quaternionic windows is unsupported.
[Section 2, Theorem 1 and the definition of M_g in Section 1] Condition 2 of Theorem 1 asserts an operator inequality for the bi-infinite matrices ∑_l M_{g_l}(k)M_{g_l}^t(k) acting on ℓ²(Z²). For an arbitrary l² window, M_g(k) is not automatically a bounded operator on ℓ²(Z²); for M=N=1 it is a convolution operator with an l² kernel, which can be unbounded. The theorem needs an explicit boundedness hypothesis on these matrices, or the inequality should be formulated as a uniform quadratic-form estimate on finitely supported sequences and then extended by density. This issue affects Theorem 1 and all later matrix-characterization theorems (Theorems 2, 3, 5, 7, and 8), since the matrix inequality is the object being verified.
[Section 3, Theorem 4 proof] The proof of Theorem 4 contains a logical gap: after deriving ∑_l ‖g_l‖² = L and observing that ‖g_l‖ ≤ 1 for every l, the paper concludes that the system is an orthonormal basis. Unit-norm vectors in a Parseval frame do form an orthonormal basis, but this requires proof; one needs, for example, to show that the Gram matrix is idempotent and has unit diagonal, forcing all off-diagonal entries to vanish. Without this step, Theorem 4, and hence Lemma 4 and Theorem 6 which depend on it, are not fully established.

minor comments (4)

[Section 2, Lemma 2 proof] In the proof of Lemma 2, the line 'the fact that p2 = k2 + 2M' appears to be a typo; it should read p2 = k2 + q2M (or p2 = k2 + qM in the second coordinate).
[Section 2, Theorem 2] The display contains the broken LaTeX string '/greaterorequalslant' and the word 'Combinig' should be 'Combining'; also, in the lower-bound half of the proof, the variable f is used where h was introduced, which should be corrected for readability.
[Section 1, Remark 1] The example in Remark 1 is hard to follow because the notation q is used for a modulation index, while the vector (1/M,1/M) is also written as a quaternionic exponential argument; the computation should be rewritten with clearer indexing.
[Section 2, Theorem 1 proof] The proof of Theorem 1 invokes Lemma 5 before that lemma is stated in Section 4; the author should either state Lemma 5 earlier or include an explicit forward reference so the reader can verify the identity in equation (2).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the quaternionic Gabor frame characterization is a self-contained reduction from definitions, with no fitted parameters, predictions, or load-bearing self-citations.

full rationale

I walked the derivation chain. Theorem 1 reduces the frame inequality for G(g,L,M,N) to a pointwise, N-periodic operator inequality on the matrices M_g(k)M_g^t(k); the proof uses only the definition of the Gabor system, Lemma 1 (geometric sums), Lemma 3 (N-periodicity), and Lemma 5 (orthonormal basis of M-periodic quaternionic sequences). There are no fitted parameters, no quantities called predictions that are actually inputs, and no uniqueness or ansatz imported from the author's prior work. The only external result, the perturbation theorem from [15], is cited as a standard frame-theoretic fact and is not equivalent to any conclusion of this paper. The apparent conjugation omissions in Lemmas 2 and 6 are mathematical correctness concerns about identities for quaternionic test functions, not circularity: those lemmas are used inside proofs, but they do not presuppose the theorems they support. Theorem 1's proof goes through the block-decomposition route and is independent of Lemma 2. Thus the central claim is self-contained and no step reduces by construction to its own input.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The main results rest on the real-valued window restriction, a standard discrete-frequency ONB for M-periodic quaternionic sequences, the external perturbation theorem from [15], and an unstated boundedness condition for the matrix functions. No free parameters or invented entities are introduced.

assumptions (5)

domain assumption The real-valued window assumption: g_l in l^2(Z x Z, R) (and similarly h_l) in Lemma 2, Theorem 1, Theorem 2, Theorem 3, Proposition 4, and implicitly in Theorem 7.
Required so that g_l commutes with the quaternionic exponentials in inner-product expansions. Lemma 2 explicitly states 'the commutativity of gl with the other terms is justified by the fact that it takes real values.' Theorem 7 omits this hypothesis, which is a flaw.
ad hoc to paper The bi-infinite matrices M_g(k), and the sums sum_l M_g(k) M_g^t(k), act as bounded operators on l^2(Z^2).
Theorem 1 uses operator inequalities on these matrices for arbitrary g in l^2; boundedness is not proved or stated. For general l^2 windows the matrix need not be bounded, so this is an unstated technical condition.
standard math The M-periodic exponentials {(1/M) e^{2 pi i m1/M k1} e^{2 pi j m2/M k2}} form an orthonormal basis for l_M(Z x Z, H) (Lemma 5).
Used to justify the periodization identity (2) in Theorem 1 and Lemma 6. It is a standard discrete Fourier fact over the quaternionic imaginary units i and j.
standard math Perturbation theorem for quaternionic frames, Theorem 4.1 of [15] (Sharma and Goel).
Invoked in Lemma 7 and Theorem 8 for stability bounds. Cited from prior literature, not self-citation.
standard math l^2(Z x Z, H) is a right quaternionic Hilbert space with inner product <f,g> = sum_k f(k) conj(g(k)).
Foundational setup; standard definition used throughout the paper.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Discrete Quaternionic (Multi-window) Gabor Systems." pith.science (2026). https://pith.science/paper/KC2JQV3M

@misc{pith2026241116988,
  author       = {Pith},
  title        = {Pith review of: Discrete Quaternionic (Multi-window) Gabor Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KC2JQV3M}},
  note         = {Machine review of arXiv:2411.16988}
}

abstract

The aim of this work is to study (Multi-window) Gabor systems in the space $\ell^2(\mathbb{Z} \times \mathbb{Z}, \mathbb{H})$, denoted by $\mathcal{G}(g,L,M,N)$, and defined by: \[ \left\{ (k_1,k_2)\in \mathbb{Z}^2\mapsto e^{2\pi i \frac{m_1}{M}k_1} g_l(k - nN) e^{2\pi j \frac{m_2}{M}k_2} \right\}_{l \in \mathbb{N}_L, (m_1, m_2) \in \mathbb{N}_M^2, n \in \mathbb{Z}^2}, \] where, $L,M,N$ are positive integers, $i,j$ are the imaginary units in the quaternion algebra, and $ \{g_l\}_{l \in \mathbb{N}_L} \subset \ell^2(\mathbb{Z} \times \mathbb{Z}, \mathbb{H}) $. Special emphasis is placed on the case where the sequences $g_l$ are real-valued. The questions addressed in this work include the characterization of quaternionic Gabor systems that form frames, the characterization of those that are orthonormal bases, and the admissibility of such systems. We also explore necessary and/or sufficient conditions for Gabor frames. The issue of duality is also discussed. Furthermore, we study the stability of these systems.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

[17]

Banach J

XL., Li, YZ Zhang, Quaternionic Gabor frame characterization a nd the density theorem. Banach J. Math. Anal. 17, 64 (2023)

work page 2023
[18]

Quaternionic Subspace Gabor Frames and Th eir Duals

Li, YZ., Zhang, XL. Quaternionic Subspace Gabor Frames and Th eir Duals. Adv. Appl. Cliﬀord Algebras 34, 38 (2024). 25

work page 2024
[1]

Adler, Quaternionic Quantum Mechanics and Quantum Fields , Oxford University Press, New York, 1995

S.L. Adler, Quaternionic Quantum Mechanics and Quantum Fields , Oxford University Press, New York, 1995

work page 1995
[2]

Balan, P.G

R. Balan, P.G. Casazza, and D. Edidin, On signal reconstruction w ithout phase, Appl. Comp. Harm. Anal. 20 (2006), 345–356

work page 2006
[3]

Benedetto, A

J. Benedetto, A. Powell, and O. Yilmaz, Sigma-Delta quantization a nd ﬁnite frames, IEEE Trans. Inform. Theory 52 (2006), 1990–2005

work page 2006
[4]

Casazza, The art of frame theory, Taiwanese J

P.G. Casazza, The art of frame theory, Taiwanese J. of Math. 4 (2000), No. 2, 129–201

work page 2000
[5]

Casazza and G

P.G. Casazza and G. Kutyniok, Frames of subspaces, Wavelets, Frames and Oper- ator Theory (College Park, MD, 2003), Contemp. Math., 345, Amer. Math. Soc., Providence, RI, 2004, 87–113

work page 2003
[6]

Q. Chen, P. Dang, and T. Qian, A frame theory of Hardy spaces w ith the quater- nionic and the Cliﬀord algebra setting, Adv. Appl. Cliﬀord Algebras 27 (2017), 1073–1101

work page 2017

Show all 18 references

[7]

Christensen, An Introduction to Frames and Riesz Bases , Applied and Numerical Harmonic Analysis, Birkh¨ auser Boston, Inc., Boston, M A, 2003

O. Christensen, An Introduction to Frames and Riesz Bases , Applied and Numerical Harmonic Analysis, Birkh¨ auser Boston, Inc., Boston, M A, 2003. 24

2003
[8]

Daubechies, A

I. Daubechies, A. Grossmann, and Y. Meyer, Painless non-orth ogonal expansions, J. Math. Physics 27 (1986), 1271–1283

1986
[9]

Dun and A.C

R.J. Dun and A.C. Schaeﬀer, A class of non-harmonic Fourier serie s, Trans. Amer. Math. Soc. 72 (1952), 341–366

1952
[10]

Ghiloni, V

R. Ghiloni, V. Moretti, and A. Perotti, Continuous slice functiona l calculus in quaternionic Hilbert spaces, Rev. Math. Phys. 25 (2013), 1350006

2013
[11]

Li and Q.-F

Y.-Z. Li and Q.-F. Lian, Gabor systems on discrete periodic sets , Sci. China Ser. A,52(2009), 1639-1660

2009
[12]

Li and Q.-F

Y.-Z. Li and Q.-F. Lian, Tight Gabor sets on discrete periodic sets , Acta Appl. Math., 107 (2009), 105-119

2009
[13]

Q.-F. Lian, J. Gong and M.-H. You, Time domain characterization of multi- window Gabor systems on discrete periodic sets , Indian J. Pure Appl. Math., 44(1):47-76, February 2013

2013
[14]

Lian and Y.-Z

Q.-F. Lian and Y.-Z. Li, The duals of Gabor frames on discrete periodic sets , J. Math. Phys., 50(2009), 013534, 22pp

2009
[15]

Sharma and S

S.K. Sharma and S. Goel, Frames in quaternionic Hilbert spaces, J. Math. Phys. Anal. Geom. 15 (2019), 395–411

2019
[16]

Heath and A.J

R.W. Heath and A.J. Paulraj, Linear dispersion codes for MIMO sy stems based on frame theory, IEEE Trans. Signal Process. 50 (2002), 2429–2441

2002

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [17]

Banach J

XL., Li, YZ Zhang, Quaternionic Gabor frame characterization a nd the density theorem. Banach J. Math. Anal. 17, 64 (2023)

work page 2023

[2] [18]

Quaternionic Subspace Gabor Frames and Th eir Duals

Li, YZ., Zhang, XL. Quaternionic Subspace Gabor Frames and Th eir Duals. Adv. Appl. Cliﬀord Algebras 34, 38 (2024). 25

work page 2024

[3] [1]

Adler, Quaternionic Quantum Mechanics and Quantum Fields , Oxford University Press, New York, 1995

S.L. Adler, Quaternionic Quantum Mechanics and Quantum Fields , Oxford University Press, New York, 1995

work page 1995

[4] [2]

Balan, P.G

R. Balan, P.G. Casazza, and D. Edidin, On signal reconstruction w ithout phase, Appl. Comp. Harm. Anal. 20 (2006), 345–356

work page 2006

[5] [3]

Benedetto, A

J. Benedetto, A. Powell, and O. Yilmaz, Sigma-Delta quantization a nd ﬁnite frames, IEEE Trans. Inform. Theory 52 (2006), 1990–2005

work page 2006

[6] [4]

Casazza, The art of frame theory, Taiwanese J

P.G. Casazza, The art of frame theory, Taiwanese J. of Math. 4 (2000), No. 2, 129–201

work page 2000

[7] [5]

Casazza and G

P.G. Casazza and G. Kutyniok, Frames of subspaces, Wavelets, Frames and Oper- ator Theory (College Park, MD, 2003), Contemp. Math., 345, Amer. Math. Soc., Providence, RI, 2004, 87–113

work page 2003

[8] [6]

Q. Chen, P. Dang, and T. Qian, A frame theory of Hardy spaces w ith the quater- nionic and the Cliﬀord algebra setting, Adv. Appl. Cliﬀord Algebras 27 (2017), 1073–1101

work page 2017