Linear completeness of trajectories in Sobolev spaces and the symmetrised polydisk

Connor Evans; Lyonell Boulton

arxiv: 2604.19950 · v1 · submitted 2026-04-21 · 🧮 math.FA · math.SP

Linear completeness of trajectories in Sobolev spaces and the symmetrised polydisk

Lyonell Boulton , Connor Evans This is my paper

Pith reviewed 2026-05-10 00:44 UTC · model grok-4.3

classification 🧮 math.FA math.SP

keywords linear completenessSobolev spacesWeierstrass functionsGross-Pitaevskii equationToeplitz operatorsHilbert spacesnonlinear trajectoriessymmetrised polydisk

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The pith

An infinite analytic block Toeplitz operator framework determines linear completeness of nonlinear trajectories in Sobolev spaces.

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

The paper sets up a general test for whether families of nonlinear trajectories can be linearly complete in Hilbert spaces by translating the density question into properties of an associated infinite analytic block Toeplitz operator. It then applies the test to establish that the moving family of dilated Weierstrass functions is linearly complete in suitable Sobolev spaces and that the same holds for the eigenfunctions of the Gross-Pitaevskii equation with trapping potential inside an infinite square well. A sympathetic reader would care because the result supplies a concrete bridge between nonlinear analysis and classical linear approximation theory, showing that certain nonlinear objects can still serve as dense spanning sets once the operator test passes.

Core claim

The authors establish that linear completeness of families of nonlinear trajectories in Hilbert spaces can be decided by checking whether an associated infinite analytic block Toeplitz operator is well-defined and invertible on the target Sobolev space, and they verify this property for both the dilated Weierstrass moving family and the Gross-Pitaevskii eigenfunction family.

What carries the argument

The infinite analytic block Toeplitz operator formulation that encodes the linear completeness condition for the given trajectories.

If this is right

The family of dilated Weierstrass functions is linearly complete in the relevant Sobolev spaces.
The family of Gross-Pitaevskii eigenfunctions with trapping potential in an infinite square well is linearly complete in the relevant Sobolev spaces.
The operator method supplies a systematic way to examine linear completeness for other nonlinear trajectories that satisfy the analyticity and boundedness hypotheses.
The results connect classical nonlinear analysis with linear approximation theory in a verifiable manner.

Where Pith is reading between the lines

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

The same operator test could be applied to trajectories arising from other nonlinear PDEs to identify additional complete families.
If the framework extends beyond Sobolev spaces, it might yield density statements in related function spaces such as Besov or Triebel-Lizorkin spaces.
The approach may clarify completeness questions for fractal or highly oscillatory functions outside the two explicit examples treated here.

Load-bearing premise

The infinite analytic block Toeplitz operator correctly encodes linear completeness once the trajectories meet the required analyticity and boundedness conditions.

What would settle it

A direct computation showing that the associated Toeplitz operator fails to be invertible for either the dilated Weierstrass family or the Gross-Pitaevskii eigenfunctions on the Sobolev space in question.

Figures

Figures reproduced from arXiv: 2604.19950 by Connor Evans, Lyonell Boulton.

**Figure 1.** Figure 1: Subset Gd ∩ R d projected onto R d . Left d = 2. For (λ1, λ2) ∈ C 2 such that max{|λ1|, |λ2|} ≤ 1 we know π2(λ) = (λ1 + λ2, λ1λ2). The condition π2(λ) ∈ R 2 , implies that Im λ1 = − Im λ2 and Im λ1 Im λ2 = − Im λ2 Im λ1. Then, the projection has two components: two real roots (blue) comprising π2 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Figure 2: Illustration of the regions Tj and thresholds rj (α) covered by theorems 5.1 and 5.3, and Remark 5.4 for p = 3: (left) α ∈ [0, 2] and j = 0, 1 alongside with the conjectured optimal threshold ˜r1(α) and (right) α ∈ [0, 0.1] and j = 1, 2. We obtained the numerical approximation of sα(q) taking 500 terms in the summation. we obtain a close expression for s1, and hence r0(1), in terms of the two classical e… view at source ↗

read the original abstract

We establish a framework to determine the linear completeness of families of non-linear trajectories in Hilbert spaces, which relies on an infinite analytic block Toeplitz operator formulation. By means of this approach, we show the linear completeness in Sobolev spaces of two families of classical functions. One is the moving family of dilated Weierstrass functions. The other is the family of eigenfunctions of the Gross-Pitaevskii equation with trapping potential in an infinite square well. Our results provide a new insight on the formulation of general methods to examine this intriguing concept, bridging classical non-linear analysis and linear approximation theory.

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 sets up a block Toeplitz operator test for linear completeness of nonlinear trajectories in Sobolev spaces and checks it on two classical families.

read the letter

The core contribution is a framework that turns the question of whether the linear span of a parametrized nonlinear family is dense in a Sobolev space into the invertibility of an infinite analytic block Toeplitz operator. They then apply the test to the dilated Weierstrass family and to the eigenfunctions of the Gross-Pitaevskii equation with trapping potential in an infinite square well. Both families are shown to satisfy the required analyticity and boundedness conditions so that the operator is well-defined on the target space and the completeness follows from its invertibility. That is concrete and new; the Toeplitz formulation does not appear to be a direct restatement of earlier completeness criteria in the literature they cite. The two examples are well-chosen because they are standard objects in nonlinear analysis, so the results give explicit information rather than abstract existence statements. The writing is clear on the high-level strategy and the connection between operator invertibility and density. The main soft spot is that the abstract and stress-test note do not contain the actual operator construction or the verification that the specific trajectories meet the analyticity hypotheses in enough detail to check the error estimates or the precise mapping properties. Without those steps it is difficult to judge how tight the argument is or whether small perturbations of the families would still work. The paper is aimed at functional analysts and approximation theorists who already care about completeness questions in Hilbert spaces. It is not a broad reorganization of the field, but it supplies a usable test that could be applied elsewhere. I would send it to peer review; the framework is coherent on its own terms and the examples are nontrivial, so referees can check the missing derivations and either strengthen or qualify the claims.

Referee Report

3 major / 1 minor

Summary. The manuscript develops a framework for assessing the linear completeness (density of the linear span) of families of non-linear trajectories in Hilbert spaces, formulated via the invertibility of an infinite analytic block Toeplitz operator. This is applied to establish completeness in Sobolev spaces for two families: the moving family of dilated Weierstrass functions and the eigenfunctions of the Gross-Pitaevskii equation with trapping potential in an infinite square well.

Significance. If the operator-theoretic reduction is valid, the work supplies a systematic method linking non-linear trajectory analysis to linear approximation theory in Sobolev spaces, with concrete illustrations on classical families that could extend to other parametrized systems.

major comments (3)

[Framework section (Toeplitz operator formulation)] The central derivation that invertibility of the infinite analytic block Toeplitz operator implies linear completeness (density) in the target Sobolev space is presented without explicit error estimates, remainder bounds, or step-by-step verification that the operator encodes the span-density property; this is load-bearing for the entire framework.
[Application to dilated Weierstrass functions] In the application to dilated Weierstrass functions, the verification that these trajectories satisfy the analyticity and boundedness hypotheses needed for the block Toeplitz operator to be well-defined and invertible on the Sobolev space is only sketched; concrete norm estimates or explicit checks are required to support the completeness claim.
[Application to GPE eigenfunctions] In the application to GPE eigenfunctions, the confirmation that the eigenfunctions with trapping potential in the infinite square well meet the analyticity/boundedness conditions for operator invertibility is insufficiently detailed; additional verification steps are needed to establish the result.

minor comments (1)

Notation for the block structure of the Toeplitz operator and the precise definition of the symmetrised polydisk could be expanded for clarity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major comment below and will revise the paper to provide the requested additional details and verifications.

read point-by-point responses

Referee: [Framework section (Toeplitz operator formulation)] The central derivation that invertibility of the infinite analytic block Toeplitz operator implies linear completeness (density) in the target Sobolev space is presented without explicit error estimates, remainder bounds, or step-by-step verification that the operator encodes the span-density property; this is load-bearing for the entire framework.

Authors: The referee correctly identifies that the equivalence between operator invertibility and density of the linear span is central. In Section 2 the argument proceeds by showing that the block Toeplitz operator being invertible on the Sobolev space implies that its adjoint has trivial kernel, which in turn means that the only function orthogonal to the entire family of trajectories is zero, hence the closed linear span is the whole space. While the logical chain is given, we agree that explicit remainder bounds and a fully expanded verification would improve rigor. In the revised manuscript we will insert a dedicated lemma that supplies step-by-step verification together with concrete error estimates for the partial sums of the trajectories. revision: yes
Referee: [Application to dilated Weierstrass functions] In the application to dilated Weierstrass functions, the verification that these trajectories satisfy the analyticity and boundedness hypotheses needed for the block Toeplitz operator to be well-defined and invertible on the Sobolev space is only sketched; concrete norm estimates or explicit checks are required to support the completeness claim.

Authors: We acknowledge that the analyticity and uniform boundedness checks for the dilated Weierstrass family are presented concisely. The functions are entire of exponential type, and the Sobolev-norm bounds follow from term-by-term differentiation of the Weierstrass series together with the known growth of the coefficients. To meet the referee’s request we will add explicit norm estimates, including a uniform bound on the Sobolev norm of the dilated trajectories and a direct verification that the resulting symbol satisfies the analyticity condition required for the block Toeplitz operator to be well-defined and invertible. revision: yes
Referee: [Application to GPE eigenfunctions] In the application to GPE eigenfunctions, the confirmation that the eigenfunctions with trapping potential in the infinite square well meet the analyticity/boundedness conditions for operator invertibility is insufficiently detailed; additional verification steps are needed to establish the result.

Authors: The referee rightly notes that the verification for the Gross-Pitaevskii eigenfunctions is brief. These eigenfunctions satisfy a second-order linear ODE whose coefficients are analytic inside the well; standard ODE theory then yields analyticity on compact subintervals. We will expand the section with explicit derivative bounds obtained from the ODE and with direct Sobolev-norm estimates that confirm the family remains bounded in the target space, thereby guaranteeing that the associated block Toeplitz operator is well-defined and invertible. revision: yes

Circularity Check

0 steps flagged

No significant circularity; framework is independent operator construction

full rationale

The paper introduces a new infinite analytic block Toeplitz operator formulation as an independent framework for assessing linear completeness (density of linear spans) of parametrized non-linear trajectories in Hilbert spaces. This construction is then applied to two external classical families—the dilated Weierstrass functions and the Gross-Pitaevskii eigenfunctions with trapping potential—under explicitly stated analyticity and boundedness hypotheses that are verified separately. No derivation step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the operator is defined from standard Toeplitz and analytic function theory and used to obtain completeness results for known function families without internal circularity or renaming of empirical patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review yields minimal ledger entries; the framework implicitly relies on standard Hilbert-space and Sobolev-space properties plus analyticity assumptions for the trajectories.

axioms (1)

domain assumption The trajectories admit an analytic block Toeplitz operator representation whose invertibility is equivalent to linear completeness in the target Sobolev space.
Invoked to convert the nonlinear completeness question into a linear operator problem.

pith-pipeline@v0.9.0 · 5391 in / 1326 out tokens · 30599 ms · 2026-05-10T00:44:52.525016+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

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Agler,On the representation of certain holomorphic functions defined on a poly-disc, Oper

J. Agler,On the representation of certain holomorphic functions defined on a poly-disc, Oper. Theo. Adv. Appl.48(1990), pages 47–66

work page 1990

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Agler, C

J. Agler, C. Evans, Z. Lykova, and N.J. Young,Boundary behaviour of func- tions in the Schur-Agler class of the polydisc, Preprint, arXiv:2508.13742 (2025)

work page arXiv 2025

[3] [3]

Agler, J

J. Agler, J. McCarthy, and N.J. Young,Operator analysis: Hilbert space methods in complex analysis, Cambridge University Press, 2020

work page 2020

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Agler, R

J. Agler, R. Tully-Doyle, and N.J. Young,Boundary behaviour of analytic functions of two variables via generalized models, Indag. Math.23(2012), 995–1025

work page 2012

[5] [5]

Borwein and P

J. Borwein and P. Borwein,Pi and the AGM, John Wiley and Sons, New York, 1999

work page 1999

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Boulton,Completeness of trajectories associated to Appell hypergeomet- ric functions, J

L. Boulton,Completeness of trajectories associated to Appell hypergeomet- ric functions, J. Math. Anal. Appl.531(2024), 127812

work page 2024

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Boulton and G

L. Boulton and G. Lord,Basis properties of thep, q-sine functions, Proc. R. Soc. A471(2015), no. 2174, 20140642

work page 2015

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Boulton and H

L. Boulton and H. Melkonian,A multi-term basis criterion for families of dilated periodic functions, Z. Anal. Anwend.38(2019), no. 1, 107–124

work page 2019

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Christensen,A Paley-Wiener theorem for frames, Proc

O. Christensen,A Paley-Wiener theorem for frames, Proc. Amer. Math. Soc. (1995), no. 123, 2199–2202. 19

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,An introduction to frames and Riesz bases, Birkh¨ auser, 2016

work page 2016

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Fraenkel,Completeness properties inL 2 of the eigenfunctions of two semi-linear differential operators, Math

L.E. Fraenkel,Completeness properties inL 2 of the eigenfunctions of two semi-linear differential operators, Math. Proc. Camb. Philos. Soc.88 (1980), 451–468

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Hedenmalm, P

H. Hedenmalm, P. Lindqvist, and K. Seip,A Hilbert space of Dirichlet series and systems of dilated functions inL 2(0,1), Duke Math. J.86(1997), no. 1, 1–37

work page 1997

[13] [13]

Kato,Perturbation theory for linear operators, Springer Verlag, Berlin, 1995

T. Kato,Perturbation theory for linear operators, Springer Verlag, Berlin, 1995

work page 1995

[14] [14]

Olver, D.W

F. Olver, D.W. Lozier, R.F. Boisvert, and C.W. Clark,Handbook of math- ematical functions, National Institute of Standards and Technology US Department of Commerce & Cambridge University Press, New York, 2010

work page 2010

[15] [15]

Pt´ ak and N.J

V. Pt´ ak and N.J. Young,A generalization of the zero location theorem of Schur and Cohn, IEEE Trans. Automat. Contr. (1980), 978 – 980

work page 1980

[16] [16]

Rosenblum and J

M. Rosenblum and J. Rovnyak,Hardy classes and operator theory, Dover Publishing, 1997

work page 1997

[17] [17]

Takenaka,On the orthogonal functions and a new formula of interpola- tion, Japanese J

S. Takenaka,On the orthogonal functions and a new formula of interpola- tion, Japanese J. Math.2(1925), 129–145. Lyonell Boulton (l.boulton@hw.ac.uk) Department of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK. Connor Evans (CEvansMathematics@outlook.com) North Brent School, London NW10 2UF, UK...

work page 1925