How large are the gaps in phase space?

Michael Speckbacher

REVIEW 1 major objections 6 minor 24 references

How large are the gaps in phase space?

T0 review · 1 major / 6 minor · reviewed 2026-08-08 · deepseek-v4-flash

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

Pith's one-line read The paper proves a quantitative necessary condition for sampling measures: for Laguerre wavelets, every pseudohyperbolic ball that avoids the support has radius at most $1-(C_{n,\alpha}\pi A/B)^{1/\alpha}$, and for Hermite-window STFT at…

desk verdict The sector-partition proof is sound, but Theorem 3's stated bound doesn't follow from it; the constants need correcting. read the letter →

arxiv 2502.04150 v1 pith:JMM7L2V6 submitted 2025-02-06 math.FA

Michael Speckbacher This is my paper · ORCID

classification math.FA MSC 42C1542C4030H20

keywords samplingmeasureswavelettransformLaguerrewaveletsphasespacegapsBergmanspacesshort-timeFourierHermitefunctionspseudohyperbolicmetric

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

The paper asks how large an empty region in phase space can be when a measure still suffices to sample the wavelet transform. For the family of Laguerre wavelets, it proves that any pseudohyperbolic ball carrying zero sampling measure has radius at most $1 - (C_{n,\alpha}\pi A/B)^{1/\alpha}$, where $A,B$ are the sampling constants and $C_{n,\alpha}$ is an explicit constant with different values for $n=0$ and $n\in\mathbb{N}$. The same idea gives a Euclidean-radius bound $R^2 \le (2/\pi)\log(4^n 5 B/A)$ for the short-time Fourier transform with Hermite windows. The central point is that the bound depends only on the condition number $A/B$, not on how the sampling measure is arranged, and the proof is elementary, relying on explicit inner-product formulas rather than on the structure of the measure.

What carries the argument

The argument is carried by the squared ratio $H_n^\alpha(u,v) = |\langle \pi(T(0))\psi_0^\alpha, \pi(T(u))\psi_n^\alpha\rangle / \langle \pi(T(v))\psi_0^\alpha, \pi(T(u))\psi_n^\alpha\rangle|^2$, which is rotationally invariant and can be bounded explicitly on sectors of the unit disk through Lemma 2. The proof partitions the unit disk, after the Möbius map $T(u)=i(1+u)/(1-u)$, into $K$ circular sectors, places one comparison point $v_k$ in each sector, and uses the sector bound on $H_n^\alpha$ to convert the sampling lower bound $A$ into an inequality against $B$ times a constant. For the STFT the analogous ratio is $H_n(z,w) = |z|^{2n}|z-w|^{-2n}e^{-\pi(|z|^2-|z-w|^2)}$, built from the explicit Hermite inner product formula $|V_{h_n}h_0(z)| = C_n |z|^n e^{-\pi |z|^2/2}$, and the same sector-partition method yields Theorem 7.

What would settle it

For $n=0,\alpha=1$, the main theorem says $R \le 1 - \pi A/(16B)$. A concrete test would be to take a discrete set $\Lambda \subset \mathbb{C}_+$ whose support avoids the pseudohyperbolic disk $D_{0.99}(i)$, compute the frame bounds $A,B$ for the wavelet $\psi_0^1$, and check whether $A/B > 16(1-0.99)/\pi = 0.0509$; if any such frame exists, the bound fails.

Watch

Extended reading notes

Core claim

The central claim is Theorem 3: if a sampling measure for the wavelet transform with wavelet $\psi_n^\alpha$ has an empty pseudohyperbolic disk of radius $R$, then $R$ cannot exceed $1 - (C_{n,\alpha}\pi A/B)^{1/\alpha}$, with $C_{n,\alpha}=4^{-(\alpha+1)}$ for $n=0$ and $C_{n,\alpha}=6^{-(2n+\alpha+1)}$ for $n\in\mathbb{N}$. This is a quantitative necessary condition for sampling: it says that the support of every such sampling measure must meet every pseudohyperbolic disk of radius larger than that bound. The same type of result is obtained for the STFT with Hermite windows, giving $R^2 \le (2/\pi)\log(4^n 5 B/A)$ for any Euclidean disk that avoids the support. As a corollary, the wavelet result transfers to Bergman spaces on the upper half-plane, where stable sampling sets must have no pseudohyperbolic gaps beyond the stated radius.

Load-bearing premise

The proof of the wavelet bound rests on a quoted formula, not proved in this paper, for the inner products of the Laguerre wavelets; if that formula were wrong or failed for the allowed parameters, the gap bound would not follow.

Editorial extensions

If this is right

For Laguerre wavelet sampling measures, every pseudohyperbolic disk of radius larger than $1 - (C_{n,\alpha}\pi A/B)^{1/\alpha}$ must intersect the support of the measure.
For Bergman spaces on the upper half-plane, any stable sampling set has the same no-large-gap property with the explicit constant $4^{-(\alpha+1)}$.
For the STFT with Hermite windows, any sampling measure must meet every Euclidean disk of radius $R$ satisfying $R^2 > (2/\pi)\log(4^n 5 B/A)$.
Because the estimates are independent of the structure of the measure, they apply uniformly to discrete frames and to continuous sampling measures.
The bounds are monotone in the condition number: a sampling system with $A$ close to $B$ forces every pseudohyperbolic disk above a small radius to be hit, while a system with large $B/A$ can tolerate larger empty regions.

Reading between the lines

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

The same sector-partition and ratio-bound scheme could be applied to other admissible wavelets with explicit transform formulas, such as Cauchy or hypergeometric wavelets, yielding gap bounds with constants determined by the corresponding inner-product formulas.
The necessary condition in Theorem 3 might combine with existing lower-density or Beurling-density results for wavelet frames to give a sharper two-sided characterization of sampling sets, since it constrains holes from above while density theorems constrain sparsity from below.
For the STFT with Hermite windows, the logarithmic dependence on $B/A$ suggests that the maximal gap radius grows only slowly as sampling becomes ill-conditioned; testing with structured Gabor lattices could show whether the $\log(B/A)$ rate is optimal.

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

Hi [Colleague],

Quick take on Speckbacher's 'How large are the gaps in phase space?': the sector-partition idea is genuinely new, but the main theorem as printed is not what the proof actually shows. The proof gives a valid quantitative bound; the theorem states a different, stronger bound.

What's good: The adaptation of Iosevich-Pedersen to the ax+b group is a real advance, and the proof is elementary and mostly self-contained. I checked Lemma 2 and the sector argument; they hold. For n=0, the proof derives R ≤ 1 - (A/(π 4^{α+1} B))^{1/α}. For n≥1, the proof's arithmetic leads to a denominator of 4π 18^n 6^α. But Theorem 3 states R ≤ 1 - (C_{n,α} π A/B)^{1/α} with C_{n,α}=4^{-(α+1)} (n=0) and 6^{-(2n+α+1)} (n≥1). That puts π in the numerator and uses 6^{2n} instead of 18^n. These don't match the proof. It looks like an algebraic slip in the theorem statement, not a flaw in the method. Still, it's load-bearing: the headline result as written is unproven.

Minor things: the notation clash between pseudohyperbolic D_R(z) and Euclidean D_R is confusing; Theorem 7's proof indexes six points for five sectors (k=0,...,5 should be 0,...,4). The STFT section is a fine self-contained presentation of a bound the author credits to Papageorgiou–van Velthoven; nothing wrong there.

The reliance on Proposition 1 from [6] is acceptable—it's a standard formula and not circular—though a referee may want it proved in an appendix.

Bottom line: the paper has a new result worth publishing once the theorem statement is corrected to match the proof. This is exactly what a referee should catch. I'd send it out, with a request to verify all constants carefully. If the author fixes the statement, it's a solid contribution. If they can sharpen the proof to the stated bound, even better.

Recommendation: send to peer review, but expect major revision.

Best,

[You]

Referee Report

1 major / 6 minor

Summary. The paper studies sampling measures for the wavelet transform with Laguerre wavelets and for the short-time Fourier transform with Hermite windows, and proves quantitative upper bounds on the radius of a ball (pseudohyperbolic for the continuous wavelet transform, Euclidean for the STFT) that is disjoint from the support of the sampling measure. The bounds depend only on the condition number A/B of the sampling constants and on the parameters (n, alpha). The proofs are elementary and rely on explicit inner-product formulas, a sector-partition argument, and a comparison of the sampling constants A and B. A corollary for weighted Bergman spaces is also stated.

Significance. If the main result is correct, the paper makes a valuable contribution to the theory of sampling measures for the wavelet transform, where quantitative relative-density bounds are rare. The proof strategy is elementary, gives explicit constants, and the STFT version is consistent with the existing literature. However, the central theorem as stated is not supported by the proof in the n in N case because the constants in the theorem and in the proof do not match. This undermines the main claim as written, although the general approach appears sound and likely fixable.

major comments (1)

[Theorem 3 (Section II.C)] The statement of Theorem 3 for n in N gives R <= 1 - (pi A / (6^{2n+alpha+1} B))^{1/alpha}, but the proof concludes with R <= 1 - (A / (4 pi 6^{2n+alpha} B))^{1/alpha}. These two inequalities are not equivalent: the inner fraction in the theorem is pi / 6^{2n+alpha+1}, while the proof's inner fraction is 1 / (4 pi 6^{2n+alpha}), and the former is larger by a factor of 4 pi^2 / 6 ≈ 6.58. Therefore, the theorem claims a strictly stronger bound than the proof establishes. Moreover, the intermediate estimate obtained in the proof, A <= pi 2^{n+alpha+2} 3^{2n+alpha} B (1-R)^alpha, is itself insufficient to yield the stated bound for n = 1, 2 (for n >= 3 it is actually stronger than the statement, but the proof's final line still uses the weaker constant 4 pi 6^{2n+alpha}). The theorem statement must be corrected to match the proof, or the proof must be sharpened to justify the claimed constant.

minor comments (6)

[Section II.B] The symbol D_R is used both for pseudohyperbolic disks in C+ and for Euclidean disks in the unit disk D. In the proof of Theorem 3, D_R in expressions like S_k \ D_R refers to the Euclidean disk, while in the theorem statement D_R(z) is pseudohyperbolic. This double use is confusing; consider denoting the Euclidean disk by another symbol, for example Delta_R or E_R.
[Theorem 3 proof] The sum in the display after the sector partition is written as sum_{k=1}^K, but the sectors are indexed k = 0, ..., K-1, and the final step uses v_0. Please make the indexing consistent, for instance by summing from k = 0 to K-1.
[Theorem 7 proof] The points u_k are listed for k = 0, 1, .., 5, but there are only five sectors; likely k should run from 0 to 4. In addition, the sum is written as k = 1 to 5 without specifying how the sectors are indexed; please clarify the correspondence between the summands and the sectors S_0, ..., S_4.
[Theorem 7 proof] The equality |<pi(u_k) h0, pi(w-z) h_n>| = |<pi(z+u_k) h0, pi(w) h_n>| is used implicitly when passing to the integral over C. This follows from the covariance property (8), but it would improve readability to state it explicitly.
[Remark 4(3)] The remark claims that the approach can be adapted to Lp-sampling measures, but no precise statement or proof is given. Either expand this remark with a concrete formulation or remove it to avoid an unsupported assertion.
[Proposition 1 and references] Proposition 1 is quoted from [6], a paper that shares an author with the present manuscript. The formula is explicit and standard, but since it is the only externally cited ingredient, a short verification or a more standard textbook reference would make the paper fully self-contained and remove any concern about reliance on a closely related source.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main gap bound is derived from explicit, parameter-free inner-product formulas and elementary estimates, not from the target conclusion.

full rationale

The derivation chain is self-contained. Theorem 3 reduces the sampling inequality A||ψ||^2 ≤ ∫ |⟨π(z)ψ_0, π(w)ψ_n⟩|^2 dμ(w) to a sum over circular sectors; the factor H_n^α(u,v) is computed exactly from the Laguerre-wavelet inner-product formula quoted in Proposition 1. Proposition 1 is an explicit identity cited from [6], a paper sharing an author, but it is parameter-free with stated assumptions, does not contain the target gap bound, and is independently checkable; citing it is therefore real evidence, not circular input. Lemma 2 and the sector partition then produce a numerical constant, and the final bound R ≤ 1 − (C_{n,α} π A/B)^{1/α} follows by solving the resulting inequality. The limit κ→1 for n=0 is justified by continuity of the right-hand side. The STFT proof (Theorem 7) uses the standard Hermite identity |V_{h_n h_0}(z)| = C_n |z|^n e^{−π|z|^2/2} together with the covariance formula (8) and Lemma 6; again no fitted parameter or target-equivalent input appears. No step renames a known result or imports a uniqueness theorem. The only self-citation is the Laguerre inner-product formula, which is load-bearing for the proof but not circular because it is an independent, explicit computational identity.

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

The paper introduces no free parameters, fitted constants, or invented physical or mathematical entities. All constants in the bounds are explicit functions of A, B, n, and alpha. The derivation relies on established formulas for Laguerre and Hermite functions, with one formula quoted from the author's prior work [6].

assumptions (5)

domain assumption Explicit inner-product formula for Laguerre wavelets (Proposition 1, cited from [6, Proposition 1])
The sector estimates in Theorem 3 hinge on this formula being correct for all admissible alpha, n, and points in C+. It is stated without proof in the present paper and is a known explicit computation in the Laguerre wavelet literature.
domain assumption The functions {psi_n^alpha}_{n in N0} form an orthonormal basis of admissible wavelets in H2(C+)
Used to set the norm of pi(z)psi_0^alpha equal to 1 and to justify using the sampling inequality with W_{psi_n^alpha}; standard in the cited Laguerre wavelet literature.
domain assumption The Bergman transform B_alpha maps H2(C+) isometrically up to a constant onto A_alpha(C+)
Used in Corollary 5 to transfer the wavelet sampling bound to weighted Bergman spaces; the paper states this as a known property.
standard math Covariance property and Moyal identity for the short-time Fourier transform, and the Hermite cross-ambiguity formula |V_{h_n h_0}(z)| = C_n |z|^n e^{-pi |z|^2/2}
These are standard facts in time-frequency analysis and are the basis for the STFT gap estimate in Theorem 7.
standard math Basic properties of the pseudohyperbolic metric on C+ and D, including the Mobius map and rotational invariance
These facts define the geometry of the gap balls and support the sector partition used in the proofs.

how reviews work

0 comments

Cite this review

Pith. "Pith review of How large are the gaps in phase space?." pith.science (2026). https://pith.science/paper/JMM7L2V6

@misc{pith2026250204150,
  author       = {Pith},
  title        = {Pith review of: How large are the gaps in phase space?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JMM7L2V6}},
  note         = {Machine review of arXiv:2502.04150}
}

read the original abstract

Given a sampling measure for the wavelet transform (resp. the short-time Fourier transform) with the wavelet (resp. window) being chosen from the family of Laguerre (resp. Hermite) functions, we provide quantitative upper bounds on the radius of any ball that does not intersect the support of the measure. The estimates depend on the condition number, i.e., the ratio of the sampling constants, but are independent of the structure of the measure. Our proofs are completely elementary and rely on explicit formulas for the respective transforms.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

[6]

L. D. Abreu and M. Speckbacher. Donoho-Logan large sieve principles for the wavelet transform. Appl. Comput. Harmon. Anal. , 74:101709, 2025

work page 2025
[1]

L. D. Abreu, P . Balazs, M. de Gosson, and Z. Mouayn. Discre te coherent states for higher Landau levels. Ann. Phys. , 363:337–353, 2015

work page 2015
[2]

L. D. Abreu and M. D¨ orﬂer. An inverse problem for localiz ation operators. Inverse Problems, 28(11):115001, 2012

work page 2012
[3]

L. D. Abreu and K. Gr¨ ochenig. Banach Gabor frames with He rmite functions: polyanalytic spaces from the Heisenberg group. Appl. Anal. , 91:1981–1997, 2012

work page 1981
[4]

L. D. Abreu and M. Speckbacher. Donoho-Logan large sieve principles for modulation and polyanalytic Fock spaces. Bull. Sci. Math. , 171, 2021

work page 2021
[5]

L. D. Abreu and M. Speckbacher. Afﬁne density, von Neuman n dimension and a problem of Perelomov. Adv. Math., 408:108564, 2022

work page 2022
[7]

G. Ascensi. Sampling measures for the Gabor transform. J. Approx. Theory, 200:40–67, 2015

work page 2015
[8]

E. Berge. A primer on coorbit theory. J. F ourier Anal. Appl., 28, 2022

work page 2022

Show all 24 references

[9]

Gr¨ ochenig

K. Gr¨ ochenig. Foundations of Time-Frequency Analysis . Appl. Numer. Harmon. Anal. Birkh¨ auser Boston, 2001

2001
[10]

Hartmann, D

A. Hartmann, D. Kamissoko, S. Konate, and M. A. Orsoni. D ominating sets in Bergman spaces and sampling constants. J. Math. Anal. Appl. , 495(2):124755, 2021

2021
[11]

C. Heil. History and evolution of the density theorem fo r Gabor frames. J. F ourier Anal. Appl. , 13:113–166, 2007

2007
[12]

Iosevich and S

A. Iosevich and S. Pedersen. How large are the spectral g aps? Pac. J. Math., 192(2):307–314, 2000

2000
[13]

Jaming and M

P . Jaming and M. Speckbacher. Planar sampling sets for t he short-time Fourier transform. Constr . Approx., 53:479–502, 2021

2021
[14]

Janson, J

S. Janson, J. Peetre, and R. Rochberg. Hankel forms and t he Fock space. Rev. Math. Iberoamericana , 3(1):61–138, 1987

1987
[15]

Kutyniok

G. Kutyniok. Afﬁne Density in Wavelet Analysis , volume 1914 of Lecture Notes in Mathematics . Springer V erlag, Berlin, 2007

1914
[16]

D. H. Luecking. Sampling measures for the Bergman space on the unit disk. Math. Annalen , 316(4):659–679, 2000

2000
[17]

F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark. NIST Handbook of Mathematical Functions . Cambridge University Press, 1st edition, 2010

2010
[18]

Ortega-Cerd´ a

J. Ortega-Cerd´ a. Sampling measures. Publ. Mat. , 42:559–566, 1998

1998
[19]

Papageorgiou and J

E. Papageorgiou and J. T. van V elthoven. Counting funct ion estimates for coherent frames and Riesz sequences. arXiv:2407.13540, 2024

2024 arXiv
[20]

J. L. Romero and J. T. van V elthoven. The density theorem for discrete series representations restricted to lattices. Expo. Math., 40(2):265–301, 2022

2022
[21]

K. Seip. Reproducing formulas and double orthogonalit y in Bargmann and Bergman spaces. SIAM J. Math. Anal. , 22(3):856–876, 1991

1991
[22]

K. Seip. Beurling type density theorems in the unit disk . Invent. Math., 113:21–39, 1993

1993
[23]

K. Seip. Regular sets of sampling and interpolation for weighted Bergman spaces. Proc. Amer . Math. Soc., 117:223–220, 1993

1993
[24]

Sun and X

W. Sun and X. Zhou. Irregular wavelet/Gabor frames. Appl. Comput. Harmon. Anal. , 13(1):63–76, 2002

2002

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [6]

L. D. Abreu and M. Speckbacher. Donoho-Logan large sieve principles for the wavelet transform. Appl. Comput. Harmon. Anal. , 74:101709, 2025

work page 2025

[2] [1]

L. D. Abreu, P . Balazs, M. de Gosson, and Z. Mouayn. Discre te coherent states for higher Landau levels. Ann. Phys. , 363:337–353, 2015

work page 2015

[3] [2]

L. D. Abreu and M. D¨ orﬂer. An inverse problem for localiz ation operators. Inverse Problems, 28(11):115001, 2012

work page 2012

[4] [3]

L. D. Abreu and K. Gr¨ ochenig. Banach Gabor frames with He rmite functions: polyanalytic spaces from the Heisenberg group. Appl. Anal. , 91:1981–1997, 2012

work page 1981

[5] [4]

L. D. Abreu and M. Speckbacher. Donoho-Logan large sieve principles for modulation and polyanalytic Fock spaces. Bull. Sci. Math. , 171, 2021

work page 2021

[6] [5]

L. D. Abreu and M. Speckbacher. Afﬁne density, von Neuman n dimension and a problem of Perelomov. Adv. Math., 408:108564, 2022

work page 2022

[7] [7]

G. Ascensi. Sampling measures for the Gabor transform. J. Approx. Theory, 200:40–67, 2015

work page 2015

[8] [8]

E. Berge. A primer on coorbit theory. J. F ourier Anal. Appl., 28, 2022

work page 2022