Fourier frames on smooth surfaces with nonvanishing Gaussian curvature

Bochen Liu; Xinyu Chen

REVIEW 2 major objections 5 minor 13 references

Fourier frames on smooth surfaces with nonvanishing Gaussian curvature

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A compact smooth surface with nonvanishing Gaussian curvature admits no Fourier frame.

desk verdict A genuinely new endpoint result for Fourier frames on curved surfaces, but the proof leans on an unproved observation from earlier work that needs referee scrutiny. read the letter →

arxiv 2507.05777 v1 pith:WPBBKD2L submitted 2025-07-08 math.CA math.FA

Xinyu Chen , Bochen Liu This is my paper · ORCID

classification math.CAmath.FA MSC 42C1542B10

keywords FourierframessurfacemeasureGaussiancurvatureframespectrumstationaryphasesphericalcaphemispherenormaldirections

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 compact smooth surfaces in Euclidean space with nonvanishing Gaussian curvature never admit Fourier frames, even when the surface is self-intersecting or is not the boundary of a convex body. The core of the proof is a contradiction between two consequences that any frame spectrum would force: one Fourier estimate makes a certain frequency series diverge, while another makes the same series converge. The result also settles an endpoint question from [7] by showing that a hemisphere admits no Fourier frame, placing the exact threshold between small caps, which do admit frames by [9], and larger caps. Along the way the paper shows that a spherical cap near the north pole cannot have a frame spectrum near the $x_d$-axis.

What carries the argument

The argument rests on a divergence-convergence dichotomy extracted from the earlier analysis in [6]. For a surface-carried measure $\mu$, an upper Fourier decay bound $|\widehat{\psi d\mu}(\xi)|\lesssim |\xi|^{-(d-1)/2}$ together with the lower frame inequality tested only on the exponentials $\{e^{2\pi i x\cdot\xi}\}_{\xi\in\mathbb{R}^d}$ forces the series $\sum_{\lambda\in\Lambda\setminus\{0\}}|\lambda|^{-(d-1)}$ to diverge. A lower Fourier estimate $\int_{B_1(\xi)}|\hat\mu(\eta)|^2\,d\eta\gtrsim |\xi|^{-(d-1)}$ together with the upper frame inequality tested on the same exponentials forces the same series to converge. The paper establishes both estimates on the relevant surface measures: stationary phase supplies the upper decay, and a newly localized pointwise estimate $|\widehat{\psi_{ij}\,d\sigma_i}(\lambda)|^2\gtrsim|\lambda|^{-(d-1)}$ holds for $\lambda$ along normal directions in small patches, avoiding the need to integrate over neighborhoods or to control sums of phases from multiple normal points. The divergence and convergence conclusions contradict each other.

What would settle it

Exhibit a compact smooth surface with nonvanishing Gaussian curvature and a discrete set $\Lambda$ with finite $\sum_{\lambda\ne0}|\lambda|^{-(d-1)}$ for which both frame bounds hold, and Theorem 1.1 and Corollary 1.2 would be false; more directly, test the dichotomy in [6] on a simple measure with a known frame and see whether the predicted divergence and convergence conclusions still follow.

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Extended reading notes

Core claim

Let $S$ be a compact $(d-1)$-dimensional smooth submanifold immersed in $\mathbb{R}^d$ with nonvanishing Gaussian curvature, with surface measure $\sigma_S$ normalized by its multiplicity function. Theorem 1.1 states that if $\sigma_S$ admits a frame spectrum $\Lambda$, then for every submanifold $S'$ compactly contained in $S$, the set $\Lambda$ must meet the complement of $C_{S'}$, the union of the one-dimensional normal subspaces of $S'$. For compact $S$ one may take $S'=S$, and because every direction is normal to $S$ at some point, $C_S=\mathbb{R}^d$; hence $\Lambda$ would have to contain a point outside all of $\mathbb{R}^d$, which is impossible. Therefore $\sigma_S$ admits no Fourier frame. Two consequences are proved for the sphere: a small cap near the north pole cannot have a frame spectrum near the $x_d$-axis, and any cap whose interior contains a closed hemisphere admits no Fourier frame at all. The endpoint case of a hemisphere is proved separately for the restricted surface measure $\sigma_+$ on $S\cap\{x_d\ge 0\}$ when $S$ is the smooth boundary of a centrally symmetric convex body.

Load-bearing premise

The proof depends on the dichotomy from [6] that frame inequalities tested only on exponentials force the series $\sum|\lambda|^{-(d-1)}$ to diverge under the upper Fourier decay and to converge under the lower Fourier estimate; if that dichotomy fails for these surface measures, the contradiction collapses.

Editorial extensions

If this is right

The self-intersecting planar curve $(\cos\theta+2\cos2\theta,\sin\theta+\sin2\theta)$ and its revolution surfaces in $\mathbb{R}^d$ admit no Fourier frame, even though they are not convex and their outward boundaries are not $C^3$.
For the sphere, a frame spectrum of a small cap near the north pole must avoid a neighborhood of the $x_d$-axis, quantifying the obstruction behind the known small-cap frame construction.
A spherical cap whose interior contains a closed hemisphere admits no Fourier frame; combined with the positive result for caps compactly contained in a hemisphere, the hemisphere is the exact threshold.
The hemisphere itself admits no Fourier frame, answering the question posed at the end of [7] and extending the no-frame theorem beyond the whole sphere.
The two-sided frequency-series test gives a reusable criterion: proving nonexistence of Fourier frames for a curved measure reduces to checking one upper and one lower Fourier estimate against the exponential family.

Reading between the lines

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

The pointwise lower bound along normal directions means the proof never has to compare phases from several surface points sharing the same normal, so the same strategy may extend to immersed surfaces with many-to-one normal maps, provided one point dominates in each small chart.
Compactness is used only to conclude $C_S=\mathbb{R}^d$; for a noncompact surface whose normal set is a proper cone, the theorem would predict that any frame spectrum must have frequencies outside that cone, a statement one could test directly on model surfaces.
The reduction to exponentials suggests a broader sufficient obstruction: a measure whose Fourier transform decays like $|\xi|^{-s}$ and whose local average energy is at least $|\xi|^{-2s}$ cannot carry a frame spectrum. This could be checked against other singular measures with known Fourier decay, such as self-similar measures.
Because the hemisphere proof works for any centrally symmetric smooth convex boundary, the endpoint threshold is not a special sphere phenomenon; testing non-centrally symmetric caps whose boundary is only $C^2$ would indicate how much symmetry the argument really needs.

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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

This paper genuinely advances the subject. Theorem 1.3 settles the hemisphere question asked by Kolountzakis and Lai, and Corollary 1.2 extends the no-frame theorem from [6] on convex bodies to all compact immersed surfaces with nonvanishing curvature, including self-intersecting curves. It also improves [7] from tight frames to frames. The local stationary-phase argument in Section 3 is a genuine simplification: it sidesteps the global exponential sums that made the earlier approach heavy.

I read the proofs carefully. The Morse-lemma reduction and the uniform stationary-phase estimate are standard and appear correct. The symmetry identity (2.5) is clean and does what it needs to: it transfers the lower bound (1.2) from the full surface to the half-surface, and the bump-function trick handles the boundary issue. I found no mathematical errors in the parts that are proved in the paper.

The real soft spot is the observation in Section 2. The paper needs a pair of implications from [6]: upper decay plus a lower frame bound tested only on the exponential family forces divergence of the sum, and the integrated lower bound plus an upper frame bound on exponentials forces convergence. The authors state this as an observation and do not prove it. The rest of the paper rests entirely on it, and the measures here (ψ dσ+ and σ+) have features, like the slower boundary decay in (1.8), that are not present for the whole sphere. I believe the implications are true—the exponentials are the only test functions used in [6] anyway—but a referee should verify this by checking the proofs in [6] or asking the authors to write out the argument. This is a verifiability gap, not an identified error.

Minor issues: the abstract has typos ("indclude"), and the last remark is a bit hand-wavy about why the old integral approach would be complicated, but that is harmless.

Who should read this: anyone working on Fourier frames for singular measures or exponential bases for surface measures. It deserves a serious referee. My recommendation: send it to peer review. The referee should ask the authors to make the Section 2 dependency explicit, ideally by giving a self-contained proof or stating the exact theorem from [6] that covers the exponential-family version.

Referee Report

2 major / 5 minor

Summary. The paper studies Fourier frames for surface-carried measures on smooth (d-1)-dimensional surfaces with nonvanishing Gaussian curvature. Theorem 1.1 shows that any frame spectrum for such a surface measure must intersect the complement of the normal set of every compactly contained submanifold; Corollary 1.2 concludes that compact immersed smooth submanifolds with nonvanishing Gaussian curvature admit no Fourier frames, covering self-intersecting examples beyond the convex-body case. Theorem 1.3 settles the endpoint case for hemispheres of centrally symmetric convex bodies, answering a question of Kolountzakis and Lai. The proofs use stationary phase to obtain upper Fourier decay and a local pointwise lower bound, and import from [6] an observation that the frame inequalities, tested only on the exponential family, force divergence or convergence of the series sum |lambda|^{-(d-1)}.

Significance. If the imported observation is supplied as a complete argument, the results are substantial: they generalize the main theorem of [6] from boundaries of convex bodies to immersed surfaces, improve the planar result of [7] from tight frames to general frames, and resolve the previously open hemisphere endpoint. The local argument in Section 3 is elegant and avoids the difficult exponential-sum integral that a direct approach would require. The paper is clearly written and includes explicit examples showing that the new class genuinely extends the convex setting. The main caveat is that the Section 2 observation is load-bearing and is stated without proof; this is a verifiability gap rather than an identified error.

major comments (2)

[Section 2, Eqs. (2.1)-(2.4)] The assertions that (1.1) together with the exponential-family lower inequality 1 <~ sum_lambda |hat sigma(lambda - xi)|^2 for all xi implies the divergence (2.2), and that (1.2) together with the exponential-family upper inequality implies the convergence (2.4), are stated as an observation but not proved. These assertions are load-bearing: they are used for the divergence in Theorem 1.1 and for both halves of Theorem 1.3. Please add a proof or a precise lemma statement with hypotheses, or quote the exact theorem in [6] that contains this formulation. In particular, the measures considered here, such as psi dsigma_+ and sigma_+, have features not present for the full sphere (for instance, the boundary-axis decay (1.8) for d >= 4), so it is not immediate that the hypotheses of the cited theorems in [6] hold verbatim.
[Section 3, Eq. (3.4)] The proof asserts that for each lambda in C_ij \ {0} there is a unique p_lambda in supp psi_ij with lambda as a normal. For immersed or self-intersecting surfaces this uniqueness is not automatic and needs justification; if two points in the same small support had the same normal, the pointwise stationary phase lower bound (3.5)-(3.6) would not follow as stated. Please add a short argument using nonvanishing Gaussian curvature and the choice of the Lebesgue number to show that the Gauss map is injective on each support, or restrict the supports further so that this is explicit.

minor comments (5)

[Abstract] The word 'indclude' in the abstract should read 'include'.
[Section 2, first paragraph] The phrase 'does not take use of all f' should be 'does not make use of all f'.
[Section 3, final paragraph] The phrase 'successfully avoid dealing' should be 'successfully avoids dealing' or 'successfully avoids the need to deal'.
[Reference [13]] The editor's name is given as 'Laba and Carol Shubin'; it should be 'Izabella Laba and Carol Shubin'.
[Eq. (3.4)] The stationary phase expansion gives the lower bound |widehat{psi_ij dsigma_i}(lambda)|^2 >~ |lambda|^{-(d-1)} only for sufficiently large |lambda|, not for all lambda in C_ij \ {0} as written. Please add a sentence explaining that the finitely many small elements of the discrete spectrum Lambda can be discarded or absorbed, so that the asymptotic lower bound is enough for the convergence argument.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained modulo standard stationary phase and the independent external result [6].

full rationale

The paper's main results follow from three ingredients: standard stationary-phase asymptotics; a pointwise lower bound on the Fourier transform of a localized surface measure obtained via the Morse lemma and stationary phase (Section 3, Eqs. (3.4)–(3.6)); and the divergence/convergence dichotomy for frame spectra imported from [6] (Section 2). The most citation-dependent link is the Section 2 'observation' that the proof in [6] uses only the exponential family {e^{2pi i x·xi}} rather than all f in L^2. This observation is stated without proof in the present paper and is load-bearing for both Theorem 1.1 and Theorem 1.3. However, [6] is an independent published result by Iosevich, Lai, Liu and Wyman; its assumptions concern surface-carried measures on boundaries of convex bodies and do not include the new conclusions about immersed surfaces, self-intersecting curves, or hemispheres. The present theorems are not used to prove [6], no parameter is fitted and renamed as a prediction, and no displayed equation reduces to its own input by construction. The unproved observation is a verifiability gap requiring the reader to consult [6], but it is legitimate external mathematical support rather than circularity. Consequently the circularity score is 0.

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

No free parameters, no fitted constants, and no new physical or mathematical entities are introduced. The proof relies on standard tools and on prior theorems from [6], [12], [13].

assumptions (5)

standard math Stationary phase estimates with uniform constants for oscillatory integrals over surfaces with nondegenerate phase.
Invoked in Sections 2 and 3 via Theorem 1.2.1 of [12] and Proposition 6.4 of [13].
standard math Morse lemma converts a nondegenerate critical point of the height function into a quadratic normal form.
Used in Section 3 to represent the surface locally as a graph with Hessian diagonalized.
domain assumption The implications from [6, Thm 1.3,1.4] as restated in Section 2: upper Fourier decay plus lower frame bound on exponentials implies sum |lambda|^{-(d-1)} diverges; lower integrated Fourier estimate plus upper frame bound on exponentials implies the sum converges.
This is the load-bearing bridge between Fourier bounds and frame spectra. It is cited from prior work involving one of the authors, not proved in this preprint.
domain assumption For a compact immersed surface, every direction occurs as a normal at some point, so C_S = R^d.
Support-function maximum argument in Section 1 after Theorem 1.1; requires compactness and smoothness.
domain assumption The partition of unity psi_ij can be chosen with supports so small that each normal direction appears at most once in the support.
Uses the Lebesgue number of a cover by normal-form neighborhoods in Section 3; ensures uniqueness of the stationary point in (3.5).

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Cite this review

Pith. "Pith review of Fourier frames on smooth surfaces with nonvanishing Gaussian curvature." pith.science (2026). https://pith.science/paper/WPBBKD2L

@misc{pith2026250705777,
  author       = {Pith},
  title        = {Pith review of: Fourier frames on smooth surfaces with nonvanishing Gaussian curvature},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WPBBKD2L}},
  note         = {Machine review of arXiv:2507.05777}
}

abstract

It is known that a small spherical cap (rigorously its surface measure) admits Fourier frames, while the whole sphere does not. In this paper, we prove more general results. Consequences indclude that a small spherical cap in $\mathbb{R}^d$ near the north pole cannot have a frame spectrum near the $x_d$-axis, and $S$ does not admit any Fourier frame if its interior contains a closed hemisphere. We also resolve the endpoint case, that is, a hemisphere does not admit any Fourier frame. This answers a question of Kolountzakis and Lai. Our results also hold on more general smooth surfaces with nonvanishing Gaussian curvature. In particular, any compact $(d-1)$-dimensional smooth submanifold immersed in $\mathbb{R}^d$ with nonvanishing Gaussian curvature does not admit any Fourier frame. This generalizes a previous result of Iosevich, Lai, Wyman and the second author on the boundary of convex bodies, as well as improves a recent result of Kolountzakis and Lai from tight frame to frame.

Figures

Figures reproduced from arXiv: 2507.05777 by the authors.

**Figure 1.1.** (x, y) = (cos θ + 2 cos 2θ,sin θ + sin 2θ) In R 3 , one can rotate this curve in the xy-plane about the y-axis to obtain (a(θ) cos φ, a(θ) sin φ, b(θ)), 0 ⩽ θ, φ ⩽ 2π, [PITH_FULL_IMAGE:figures/full_fig_p003_1_1.png] view at source ↗

**Figure 1.2.** It is the slice of (a(θ) cos φ, a(θ) sin φ, b(θ)) in the xy-plane. The curve that can be observed from outside is a reflection of the right half of [PITH_FULL_IMAGE:figures/full_fig_p004_1_2.png] view at source ↗

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Reference graph

Works this paper leans on

13 extracted references · 12 canonical work pages

[6]

Iosevich, C.-K

A. Iosevich, C.-K. Lai, B. Liu, and E. Wyman. Fourier frames for surface-carried measures. Int. Math. Res. Not. IMRN, (3):1644–1665, 2022

work page 2022
[7]

M. N. Kolountzakis and C.-K. Lai. Non-spectrality of some piecewise smooth curves and unions of line segments. arXiv:2507.00581

work page Pith review arXiv
[1]

R. J. Duffin and A. C. Schaeffer. A class of nonharmonic Fourier series. Trans. Amer. Math. Soc., 72:341–366, 1952

work page 1952
[2]

D. E. Dutkay and C.-K. Lai. Uniformity of measures with Fourier frames.Adv. Math., 252:684–707, 2014

work page 2014
[3]

Fu and C.-K

X. Fu and C.-K. Lai. Translational absolute continuity and Fourier frames on a sum of singular measures. J. Funct. Anal., 274(9):2477–2498, 2018

work page 2018
[4]

He, C.-K

X.-G. He, C.-K. Lai, and K.-S. Lau. Exponential spectra in L2(µ). Appl. Comput. Harmon. Anal., 34(3):327–338, 2013

work page 2013
[5]

C. S. Herz. Fourier transforms related to convex sets. Ann. of Math. (2), 75:81–92, 1962

work page 1962
[8]

C.-K. Lai. On Fourier frame of absolutely continuous measures. J. Funct. Anal., 261(10):2877– 2889, 2011

work page 2011

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[9]

N. Lev. Fourier frames for singular measures and pure type phenomena. Proc. Amer. Math. Soc., 146(7):2883–2896, 2018

2018
[10]

Li and B

L. Li and B. Liu. Fourier frames on salem measures. arXiv preprint arXiv:2506.01280

arXiv
[11]

Nitzan, A

S. Nitzan, A. Olevskii, and A. Ulanovskii. Exponential frames on unbounded sets. Proc. Amer. Math. Soc., 144(1):109–118, 2016

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[12]

C. D. Sogge. Fourier integrals in classical analysis, volume 210 of Cambridge Tracts in Mathemat- ics. Cambridge University Press, Cambridge, second edition, 2017

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[13]

T. H. Wolff. Lectures on harmonic analysis, volume 29 of University Lecture Series. American Mathematical Society, Providence, RI, 2003. With a foreword by Charles Fefferman and preface by Izabella Laba, Edited by Laba and Carol Shubin. Email address: 12432007@mail.sustech.edu...

2003

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [6]

Iosevich, C.-K

A. Iosevich, C.-K. Lai, B. Liu, and E. Wyman. Fourier frames for surface-carried measures. Int. Math. Res. Not. IMRN, (3):1644–1665, 2022

work page 2022

[2] [7]

M. N. Kolountzakis and C.-K. Lai. Non-spectrality of some piecewise smooth curves and unions of line segments. arXiv:2507.00581

work page Pith review arXiv

[3] [1]

R. J. Duffin and A. C. Schaeffer. A class of nonharmonic Fourier series. Trans. Amer. Math. Soc., 72:341–366, 1952

work page 1952

[4] [2]

D. E. Dutkay and C.-K. Lai. Uniformity of measures with Fourier frames.Adv. Math., 252:684–707, 2014

work page 2014

[5] [3]

Fu and C.-K

X. Fu and C.-K. Lai. Translational absolute continuity and Fourier frames on a sum of singular measures. J. Funct. Anal., 274(9):2477–2498, 2018

work page 2018

[6] [4]

He, C.-K

X.-G. He, C.-K. Lai, and K.-S. Lau. Exponential spectra in L2(µ). Appl. Comput. Harmon. Anal., 34(3):327–338, 2013

work page 2013

[7] [5]

C. S. Herz. Fourier transforms related to convex sets. Ann. of Math. (2), 75:81–92, 1962

work page 1962

[8] [8]

C.-K. Lai. On Fourier frame of absolutely continuous measures. J. Funct. Anal., 261(10):2877– 2889, 2011

work page 2011