Dynamical sampling, derivatives, and interpolation formulas in the Paley--Wiener space

Iker Gardeazabal-Guti\'errez; Mateus Sousa

REVIEW 2 major objections 5 minor 15 references

Dynamical sampling, derivatives, and interpolation formulas in the Paley--Wiener space

T0 review · 2 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read For N Fourier multiplier operators, an interpolation formula rebuilding every bandlimited function exists exactly when a certain N×N determinant stays bounded away from zero.

desk verdict A useful determinant criterion for interpolation formulas, probably true, but the reverse implication in Theorem 1 is not fully proved and one application section has a wrong determinant equation. read the letter →

arxiv 2506.16195 v2 pith:LDO4E2TQ submitted 2025-06-19 math.CA

Iker Gardeazabal-Gutiérrez , Mateus Sousa This is my paper · ORCID

classification math.CA MSC 42C10

keywords Paley-WienerspacesinterpolationformulasFouriermultipliersdynamicalsamplingstableframesderivativesbandlimitedfunctions

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 gives a complete criterion for when a bandlimited function can be rebuilt from N streams of samples taken through N Fourier multiplier operators $T_1,\dots,T_N$ at the decimated lattice $N\mathbb{Z}$. Its main theorem states that an interpolation formula $f(x)=\sum_{n=1}^{N}\sum_{m\in\mathbb{Z}}T_n(f)(Nm)\,g_n^T(x-Nm)$ exists for every $f$ whose Fourier transform is supported in $[-1/2,1/2]$ exactly when the matrix $M_T(x)=(N^{-1}K_n((m-1-x)/N))_{m,n=1}^{N}$ has determinant bounded away from zero almost everywhere on $((N-2)/2,N/2)$. The criterion turns an infinite system of functions into a finite matrix condition, and it comes with a norm equivalence that makes the sampling system a frame. As a direct corollary, the same determinant condition classifies stable sampling and interpolation sets, with the threshold $ ho\delta=N$ for functions of bandwidth $\delta\pi$. The proof also supplies the unique interpolating functions $g_n^T$ through the inverse matrix, recovering classical sampling formulas as special cases.

What carries the argument

The object doing the work is the $N\times N$ matrix $M_T(x)$ built from the Fourier multipliers $K_n$ sampled on the shifted lattice $(m-1-x)/N$. The first proof shows that the infinite sampling operator $A_T$ on $\ell^2$ sequences is unitarily equivalent to multiplication by $M_T(x)$ after a Fourier transform, so invertibility of $A_T$ is equivalent to the determinant of $M_T$ having a uniform positive lower bound. The second proof uses the inverse matrix directly to define the Fourier transforms of the interpolating functions. This determinant condition is the single number that controls the existence of the formula, its frame bounds, and the stable sampling/interpolation threshold.

What would settle it

Search for $N=2$ multipliers $K_1,K_2$ and densities $\delta'<\delta$ such that the sampling operator $O^{\rho,\delta'}_T$ is surjective on $PW_{\delta'\pi}$ while $O^{\rho,\delta}_T$ is injective on the larger space $PW_{\delta\pi}$; the monotonicity step asserts this cannot happen, so a concrete numerical or analytic example would refute Theorem 4's full dichotomy. For Theorem 1, the $N=1$ case already settles the mechanism: the criterion reduces to $|K_1(\xi)|\ge c>0$ on $[-1/2,1/2]$, and a multiplier with a zero, such as $K_1(\xi)=\xi$, makes constant functions invisible to the data, so no interpolation formula can exist.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1: for $N$ Fourier multiplier operators with bounded multipliers $K_n$, the sampling data $\{T_n(f)(Nm)\}_{m\in\mathbb{Z}}$ determine every $f$ in the Paley-Wiener space $PW_\pi$ through a series with translates of $N$ fixed interpolating functions if and only if $\operatorname{ess\,inf}_{x\in((N-2)/2,N/2)}|\det M_T(x)|>0$, where $M_T(x)$ is the $N\times N$ matrix whose $(m,n)$-entry is $N^{-1}K_n((m-1-x)/N)$. The interpolating functions are unique and are characterized in two equivalent ways: by the biorthogonality condition $T_m(g_n^T)(Nj)=\delta_{n,m}\delta_{j,0}$, or by the Fourier-domain identity $(\hat g_n^T((j-1-\xi)/N))_{n,j=1}^{N}=(M_T(\xi))^{-1}$. The paper also proves a norm equivalence $c_T\|f\|_2^2\le\sum_{n=1}^{N}\sum_{m\in\mathbb{Z}}|T_n(f)(Nm)|^2\le C_T\|f\|_2^2$, and proves in Theorem 4 that under the same determinant condition the pair $(\rho\mathbb{Z},T)$ is a stable sampling set for $PW_{\delta\pi}$ exactly when $\rho\delta\le N$ and an interpolation set exactly when $\rho\delta\ge N$.

Load-bearing premise

The classification of stable sampling and interpolation for all bandwidths in Theorem 4 rests on the asserted but unproved monotonicity fact that if the sampling operator is injective on a larger Paley-Wiener space, then its restriction to a smaller space is not surjective and, dually, if it is surjective on a smaller space then it is not injective on a larger one; if that fact fails for some multiplier family, the inequalities in Theorem 4 beyond the base density $\rho\delta=N$ do not follow, while Theorem 1 itself is unaffected.

Editorial extensions

If this is right

If the determinant condition holds, every $f\in PW_\pi$ is determined by the $N$ sequences $\{T_n(f)(Nm)\}_{m\in\mathbb{Z}}$, and the reconstruction series converges both in $L^2$ and uniformly on compact sets.
The reconstruction is stable: the data-to-function map is an isomorphism, so small changes in the sampled data produce small changes in the reconstructed function.
For arbitrary bandwidth, the same determinant condition implies that $(\rho\mathbb{Z},T)$ is a stable sampling set for $PW_{\delta\pi}$ exactly when $\rho\delta\le N$ and an interpolation set exactly when $\rho\delta\ge N$, with the critical case $\rho\delta=N$ corresponding to the interpolation formula.
The theorem recovers classical formulas as instances: choosing $T_n$ as derivatives yields derivative-based interpolation formulas with explicit sinc-type kernels, and choosing translations yields formulas from shifted samples.
For the special dynamical-sampling family $T_n=T^{n-1}$, the determinant factors into pairwise differences of the multiplier values, so injectivity of the multiplier on the relevant lattice points is sufficient for an interpolation formula.
If all multipliers share a common zero, no interpolation formula exists, which explains why a family of derivative operators without an independent term cannot support such a formula.

Reading between the lines

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

The determinant criterion gives a purely finite-dimensional test: for any proposed multipliers $K_n$, computing the essential infimum of $|\det M_T|$ over one interval decides whether the sampling scheme works, so one could search computationally for multiplier families with good reconstruction properties.
The unproved monotonicity step in the proof of Theorem 4 is the only load-bearing point beyond the base case; if that monotonicity fails for some multiplier family, the stable-sampling/interpolation classification for $\rho\delta\neq N$ would need a different argument, while Theorem 1 would stand.
The same matrix condition may extend to vector-valued Paley-Wiener spaces or higher-dimensional lattices, where the Poisson summation step in Lemma 6 would be replaced by the corresponding multidimensional identity; the paper does not pursue this.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Quick take: this paper gives a genuinely useful criterion for when an interpolation formula exists using samples of N Fourier multiplier operators on N Z. I think the criterion is true, but the proof as written has a real gap in the reverse implication, and Section 3.6 contains an incorrect displayed equation. It deserves a serious referee, but it needs revision before I would rely on the applications.

What is new and good: the literature mostly treats specific operators like derivatives or translations, while Theorem 1 handles N arbitrary bounded Fourier multipliers and characterizes the interpolating functions uniquely. The matrix M_T and the reduction via Poisson summation are clean. Lemma 5 and Lemma 6 are sound, and the two proofs are genuinely different routes: one through the operator A_T on sequence spaces, one through the Fourier domain. Corollary 2 and the applications recovering Vaaler's and Littmann's formulas, plus the new mixed-derivative and translation formulas, are valuable. The authors are also honest: they do not claim the self-citation to [13] does more than supply a proof technique.

Soft spots, in proportion:

1. The main gap is in the implication “interpolation formula exists” ⇒ “ess inf |det M_T| > 0.” In both proofs the argument stops with M_T^{-1} equal to a matrix of Fourier transforms of PW_pi functions, which gives L^2 membership, not the required L^\infty bound. That rules out det = 0 on a set of positive measure but does not rule out det approaching zero. The missing step is a uniform boundedness argument for the synthesis operator, or something equivalent, to force the inverse matrix into L^\infty. This is load-bearing.

2. The monotonicity step in Theorem 4 is asserted without proof. The stress-test note is right that this is an elementary restriction argument, so it is minor, but it should still be written out.

3. Section 3.6 is not reliable. The displayed equation for the vanishing determinant is not the determinant of the matrix, and the listed exceptional cases do not follow from the actual determinant. This section needs to be redone.

4. Smaller issues: Theorem 1 has typos (“k” for “N”, “3)” for “2)”), and Theorem 4 writes “ess inf M_T” where it should write “ess inf |det M_T|”.

Who this is for: people working in sampling theory, shift-invariant spaces, and Fourier interpolation. The central criterion is worth discussing even in its current state. Recommendation: conditional rejection with a clear path — the authors should supply the missing L^\infty argument and correct Section 3.6. I would send it to a serious referee rather than desk-reject.

Referee Report

2 major / 5 minor

Summary. The paper studies interpolation formulas in the Paley--Wiener space PW_π of the form f(x)=Σ_{n=1}^N Σ_{m∈Z} T_n(f)(Nm) g_n^T(x-Nm), where T_n are Fourier multiplier operators with bounded multipliers K_n. The central result, Theorem 1, characterizes the existence of such formulas by the condition ess inf |det M_T|>0, where M_T(x)=(N^{-1} K_n((m-1-x)/N))_{m,n=1}^N; it also gives two characterizations of the interpolating functions. Theorem 4 extends the criterion to stable sampling and interpolation sets of the form (ρZ,T). The paper contains two proofs of Theorem 1 and several applications, including recovery of formulas of Vaaler and Littmann, formulas with translations and derivatives, and a difference-quotient example.

Significance. If Theorem 1 is fully established, it gives a clean and useful matrix-determinant criterion that unifies several classical interpolation formulas and places them in a dynamical-sampling framework. The paper has genuine strengths: Lemma 5 and Lemma 6 are correct and efficient tools; the direction 1)=>2) of Theorem 1 is essentially complete; the recovery of the Vaaler and Littmann formulas is explicit and instructive; and the applications are concrete. However, the reverse implication 2)=>1) of Theorem 1 is underproved in both proofs, and Section 3.6 contains an incorrect determinant condition. These issues are load-bearing and need repair before the paper can be accepted.

major comments (2)

[Section 2.3 and 2.2] The implication 2)=>1) of Theorem 1 is not proved as written. In the second proof, after deriving the identity Id = M_T(ξ) · (\hat{g}_n^T((j-1-ξ)/N))_{n,j} almost everywhere, the paper concludes characterization ii). But this identity only shows that M_T(ξ)^{-1} is a matrix whose entries are restrictions of \hat{g}_n^T, hence belong to L^2; it does not imply that M_T(ξ)^{-1} is in L^∞, which is exactly what ess inf |det M_T|>0 requires. For example, for N=1 the multiplier K(ξ)=|ξ|^{1/4} has K^{-1}∈L^2 on [-1/2,1/2] while ess inf |K|=0. To complete the proof one must use the assumed L^2 and uniform convergence of the interpolation formula to show that the synthesis operators are uniformly bounded, forcing M_T^{-1}∈L^∞. The first proof in Section 2.2 has the same gap in a different form: the operator U defined by convolution with the sequences (g_s^T(Nm+n-1))_{m} is claimed to be the inverse of A_T, but boundedness of U on ℓ^2 is not established; that boundedness is equivalent to an L^∞ bound on the periodized Fourier symbol of g_s^T, again the missing bound on M_T^{-1}.
[Section 3.6] The determinant condition in Section 3.6 is stated incorrectly. For the displayed 2×2 matrix, direct computation gives determinant (πi/4)e^{-(a+b)πi x}[(1-x)e^{bπi}sinc(ϵ(1-x)) + x e^{aπi}sinc(ϵx)], so the vanishing condition is (1-x)sinc(ϵ(1-x)) + x e^{(a-b)πi}sinc(ϵx)=0, not e^{bπi(1-x)}sinc(ϵ(1-x)) + e^{aπi x}sinc(ϵx)=0 as printed. The three listed exceptional cases are consistent with the corrected equation, but the displayed equation and the denominators in the inverse matrix that follow are incorrect. Since the existence and non-existence conclusion for this family depends on this condition, the subsection needs a careful correction.

minor comments (5)

[Theorem 1] In the statement of Theorem 1, 'There exists, for n=1,...,k, functions' should read 'for n=1,...,N'.
[Corollary 2] Corollary 2 refers to 'statements 1), 2) or 3)', but Theorem 1 lists only statements 1) and 2); the numbering should be corrected.
[Section 2.3] In the second proof, the functions u_n^T are defined only on the interval (-1/2, -1/2+1/N), but they are subsequently used on all of (-1/2,1/2). The required extension should be stated explicitly.
[Theorem 4] The conditions in Theorem 4 should involve |det M_T^ρ(x)|, not M_T^ρ(x) itself; as printed, 'ess inf M_T^ρ(x)>0' is not meaningful for a matrix-valued function. Also, in the proof, 'δ=N/δ' should be 'δ=N/ρ'.
[Section 2.4] The monotonicity step used at the end of the proof of Theorem 4 is correct, but it deserves a one-sentence justification: if the restriction of an injective bounded operator to a proper closed subspace were surjective, the operator would have a bounded right inverse into that subspace, forcing the original operator to be bijective and the subspace to be the whole space.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the interpolation criterion is derived from the external Shannon–Whittaker theorem and direct Fourier-matrix arguments; the self-citation [13] is only a non-load-bearing analogy pointer.

full rationale

The paper's central result, Theorem 1, is derived rather than assumed. The first proof (Section 2.2) writes the target data {T_n(f)(Nm)} as an operator A_T applied to Shannon–Whittaker data {f(Nm+n-1)}; since Shannon–Whittaker is an external, independently established interpolation theorem, the reduction is legitimate. A_T is then conjugated by a Fourier-series isomorphism and identified as multiplication by M_T(x) up to a unimodular Vandermonde factor, so Lemma 5 (bounded inverse iff ess inf |det| > 0) closes the equivalence. The second proof (Section 2.3) constructs the interpolating functions directly from the bounded inverse matrix M_T^{-1} under condition 1), and in the converse direction derives the matrix identity M_T (g-hat matrix) = I from the assumed interpolation formula; even if that converse requires an additional L^infty/boundedness argument, that is a proof gap, not a circular definition. Section 3 recovers Vaaler's and Littmann's formulas by applying the criterion to derivative multipliers; those classical formulas are not used to prove the criterion. The only self-citation is [13] (one coauthor overlap), and it is used merely as a remark that the Shannon–Whittaker-to-A_T idea was explored there; it is not load-bearing. No fitted parameters are renamed as predictions, and no uniqueness theorem is imported from the authors' earlier work. The asserted 'easy to deduce' monotonicity step in Theorem 4 concerns restrictions of sampling operators and is again a justification issue, not an importation of the paper's own target result. Hence the derivation chain is self-contained, and no circularity is present.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted and no new entities are introduced. The paper's claims rest only on stated hypotheses (bounded Fourier multipliers, support in [-1/2,1/2]) and standard theorems of Fourier analysis and functional analysis. The determinant condition is a theorem, not an ad hoc assumption.

assumptions (4)

standard math Paley-Wiener theorem: functions in PW_{2 pi delta} are exactly entire functions of exponential type 2 pi delta whose restriction to R is in L^2(R).
Invoked in Section 1 to justify uniform convergence of interpolation formulas on compact sets and the identification of PW spaces.
standard math Shannon-Whittaker sampling theorem: {sinc(.-n)}_{n in Z} is an orthonormal basis of PW_pi and f(x)=sum_n f(n) sinc(x-n).
Used throughout the first proof of Theorem 1 to reduce the problem to inverting an operator between ell^2 spaces.
standard math Poisson summation formula: sum_{m in Z} g-hat(m) e^{2 pi i m x} = sum_{m in Z} g(x+m) for suitable g.
Used in Lemma 6 to pass from samples of T(f) to periodized frequency sums.
standard math Open mapping theorem and Lemma 5: a matrix of L^infinity functions defines a bounded invertible operator on (L^2)^N if and only if ess inf |det| > 0.
Used in Lemma 5 and Corollary 2 to convert invertibility of matrices of multipliers into frame bounds.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Dynamical sampling, derivatives, and interpolation formulas in the Paley--Wiener space." pith.science (2026). https://pith.science/paper/LDO4E2TQ

@misc{pith2026250616195,
  author       = {Pith},
  title        = {Pith review of: Dynamical sampling, derivatives, and interpolation formulas in the Paley--Wiener space},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LDO4E2TQ}},
  note         = {Machine review of arXiv:2506.16195}
}

abstract

In this paper we present a criteria to obtain interpolations formulas in terms of the sequence $\left(\{T_n(f)(Nm)\}\}_{m\in\mathbb{Z}}\right)_{n=1}^N$, where $f$ are functions whose Fourier transform is supported in $[-1/2,1/2]$, and $T_n$ are certain Fourier multiplier operators. We also discuss applications and also prove that our results recover several classical formulas.

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[13]

J. P. G. Ramos, M. Sousa Perturbed interpolation formulae and applications, Anal. PDE 16 (2023), no. 10, 2327--2384

work page 2023
[1]

Adroubi, C

A. Adroubi, C. Cabrelli, U. Molter, S. Tang, Dynamical sampling, Appl. Comput. Harmon. Anal. 42 (2017), no. 3, 378--401

work page 2017
[2]

Aldroubi, J

A. Aldroubi, J. Davis, and I. Krishta, Dynamical sampling: time-space trade-off, Appl. Comput. Harmon. Anal. 34 (2013), no. 3, 495--503

work page 2013
[3]

Cabrelli, U

C. Cabrelli, U. Molter, V. Paternostro, F. Philipp, Dynamical sampling on finite index sets, J. Anal. Math. 140 (2020), 637--667

work page 2020
[4]

Cohn, From sphere packing to Fourier interpolation Bull

H. Cohn, From sphere packing to Fourier interpolation Bull. Amer. Math. Soc. 61 (2024), 3--22

work page 2024
[5]

Cohn and F

H. Cohn and F. Gon calves , An optimal uncertainty principle in twelve dimensions via modular forms , Invent. Math. 217 (2019), no. 3, 799--831

work page 2019
[6]

H. Cohn, A. Kumar, S. Miller, D. Radchenko and M. Viazovska, The sphere packing problem in dimension 24, Ann. Math. 185 (2017), no. 3, 1017--1033

work page 2017
[7]

H. Cohn, A. Kumar, S. Miller, D. Radchenko and M. Viazovska, Universal optimality of the E_8 and Leech lattices and interpolation formulas, Ann. Math. 196 (2022), no. 3, 983--1082

work page 2022

Show all 15 references

[8]

Ghosh and A.A

R. Ghosh and A.A. Selvan, Sampling and Interpolation of Periodic Nonuniform Samples Involving Derivatives, Results Math 78, 52 (2023)

2023
[9]

Gonçalves, Interpolation formulas with derivatives in de Branges Spaces, Trans

F. Gonçalves, Interpolation formulas with derivatives in de Branges Spaces, Trans. Amer. Math. Soc. 369 (2017), 805--832

2017
[10]

Gonçalves, F

F. Gonçalves, F. Littmann, Interpolation Formulas with Derivatives in De Branges Spaces II, J. Math. Anal. Appl. 458 (2018), 1091--1114

2018
[11]

Littmann, Entire Function Majorants , PhD Thesis (2003), University of Illinois at Urbana-Champaign

F. Littmann, Entire Function Majorants , PhD Thesis (2003), University of Illinois at Urbana-Champaign

2003
[12]

Radchenko, M

D. Radchenko, M. Viazovska, Fourier interpolation on the real line, Publ. Math. Inst. Hautes \'Etudes Sci. 129 (2019), 51--81

2019
[14]

Vaaler, Some extremal functions in Fourier analysis, Bull

J. Vaaler, Some extremal functions in Fourier analysis, Bull. Amer. Math. Soc. 12 (1985), 183--215

1985
[15]

Viazovska, The sphere packing problem in dimension 8 , Ann

M. Viazovska, The sphere packing problem in dimension 8 , Ann. Math. 185 (2017), no. 3, 991--1015

2017

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [13]

J. P. G. Ramos, M. Sousa Perturbed interpolation formulae and applications, Anal. PDE 16 (2023), no. 10, 2327--2384

work page 2023

[2] [1]

Adroubi, C

A. Adroubi, C. Cabrelli, U. Molter, S. Tang, Dynamical sampling, Appl. Comput. Harmon. Anal. 42 (2017), no. 3, 378--401

work page 2017

[3] [2]

Aldroubi, J

A. Aldroubi, J. Davis, and I. Krishta, Dynamical sampling: time-space trade-off, Appl. Comput. Harmon. Anal. 34 (2013), no. 3, 495--503

work page 2013

[4] [3]

Cabrelli, U

C. Cabrelli, U. Molter, V. Paternostro, F. Philipp, Dynamical sampling on finite index sets, J. Anal. Math. 140 (2020), 637--667

work page 2020

[5] [4]

Cohn, From sphere packing to Fourier interpolation Bull

H. Cohn, From sphere packing to Fourier interpolation Bull. Amer. Math. Soc. 61 (2024), 3--22

work page 2024

[6] [5]

Cohn and F

H. Cohn and F. Gon calves , An optimal uncertainty principle in twelve dimensions via modular forms , Invent. Math. 217 (2019), no. 3, 799--831

work page 2019

[7] [6]

H. Cohn, A. Kumar, S. Miller, D. Radchenko and M. Viazovska, The sphere packing problem in dimension 24, Ann. Math. 185 (2017), no. 3, 1017--1033

work page 2017

[8] [7]

H. Cohn, A. Kumar, S. Miller, D. Radchenko and M. Viazovska, Universal optimality of the E_8 and Leech lattices and interpolation formulas, Ann. Math. 196 (2022), no. 3, 983--1082

work page 2022