On the construction of de Branges spaces for dynamical systems associated with finite Jacobi matrices

A.S. Mikhaylov; V.S. Mikhaylov

REVIEW 2 major objections 4 minor 15 references

On the construction of de Branges spaces for dynamical systems associated with finite Jacobi matrices

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that $B_N$, the Fourier image of the reachable set, is a de Branges space for any finite Jacobi matrix.

desk verdict The main theorem is true and the construction is a legitimate new application of the author's boundary-control framework, but the proof as written contains a misstated multiplier in the de Branges criterion and a smaller conjugation slip in the reproducing kernel identity; both are easily repaired. read the letter →

arxiv 2505.20905 v1 pith:AOBODQW5 submitted 2025-05-27 math.AP math.DSmath.SP

A.S. Mikhaylov , V.S. Mikhaylov This is my paper

classification math.APmath.DSmath.SP MSC 47B3646E2230H1035L0593B05

keywords deBrangesspacesboundarycontrolmethodfiniteJacobimatricesKreinequationsreproducingkernelsspectralfunctioninverseproblemscontrollability

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 shows that a second-order wave-type system governed by a finite Jacobi matrix $A$, with one boundary control, has a natural de Branges space of polynomials attached to it. The construction takes the reachable set $U_T$ of states attainable at time $T$, applies the spectral Fourier transform $\mathcal{F}$, and equips the polynomial space $B_N=\mathcal{F}U_T$ with the inner product defined by the system's connecting operator $C_T$. The authors verify the three axioms of the de Branges characterization: bounded point evaluations through an explicitly constructed reproducing kernel $J_z$, norm-invariance under $F^\#(z)=F(\bar z)$, and the division property for functions with a zero. The result matters because it extends the boundary control method for building de Branges spaces to a system that lacks exact controllability and has infinite propagation speed.

What carries the argument

Three objects carry the argument. (1) The control operator $W_T:f\mapsto u^f(T)$ and the boundary-controllability result Lemma 2: $W_T$ maps the special $N$-dimensional family $\mathcal{F}_T^1=\mathrm{Lin}\{S_k(T-t)\}_{k=1}^N$ isomorphically onto the state space $H_N$, so $U_T=W_T\mathcal{F}_T^1$ and the connecting operator $C_T=W_T^*W_T$ provides a genuine inner product on Fourier images. (2) The spectral Fourier transform $F:\mathbb{R}^N\to L^2_\rho(\mathbb{R})$, $b\mapsto\sum_k b_k\phi_k(\lambda)$, whose image of $U_T$ is $B_N$; on $B_N$ the connecting-operator metric coincides with $\int H\bar G\,d\rho$. (3) The special controls $j_z$ supplied by the Krein equations, which drive the system to the state $(\phi_1(z),\dots,\phi_N(z))$; their Fourier images are the reproducing kernels $J_z(\lambda)=\sum_k\phi_k(z)\phi_k(\lambda)$. Theorem 3 then converts these ingredients into the structural conclusion that $B_N$ is a de Branges space.

What would settle it

For a concrete finite Jacobi matrix and time $T$ (for instance $N=2$, $a_1=1$, $b_1=0$, $b_2=2$, $T=1$), compute the connecting operator $C_T$ from formula (7), solve the special control problem for two complex parameters $z,w$, and check that the inner product $(J_z,J_w)_{B_T}$ equals $\sum_{k=1}^N\phi_k(z)\phi_k(w)$. A mismatch for any $z,w$ would show the claimed reproducing kernel or metric is not the de Branges one.

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Extended reading notes

Core claim

The central claim is that $B_N=\mathcal{F}U_T=\mathrm{Lin}\{\phi_1,\dots,\phi_N\}$, the polynomials of degree at most $N-1$, equipped with the metric $(H,G)_{B_T}=(C_T h,g)_{\mathcal{F}_T}$ for $H=\mathcal{F}u^h(T)$ and $G=\mathcal{F}u^g(T)$ with $h,g\in\mathcal{F}_T^1$, is a de Branges space in the sense of Theorem 3. Equivalently, this metric is the $L^2$ inner product against the spectral measure $\rho$ of $A$, so $B_N$ is the spectral subspace $L_N$ with reproducing kernel $J_z(\lambda)=\sum_{k=1}^N\phi_k(z)\phi_k(\lambda)$. The proof checks the three conditions of the de Branges criterion: point evaluations satisfy $|G(z)|=|(J_z,G)_{B_N}|\le\|C_T^{1/2}j_z\|_{\mathcal{F}_T}\|G\|_{B_N}$; conjugation preserves the $L^2_\rho$ norm; and the quotient $((z-\omega)/(z-\omega))G(z)$ has unit modulus on $\mathbb{R}$, hence the same norm. Since all three hold, $B_N$ is a de Branges space and the associated Hermite-Biehler function $E$ exists by Theorem 3.

Load-bearing premise

The construction stands or falls on Lemma 2: every state of the system must be reachable by some control from the special $N$-dimensional family $\mathcal{F}_T^1=\mathrm{Lin}\{S_k(T-t)\}_{k=1}^N$, because only then does the connecting operator $C_T$ define a nondegenerate inner product on the Fourier images.

Editorial extensions

If this is right

The de Branges space $B_N$ is determined by inverse data alone: the connecting operator $C_T$, the subspace $\mathcal{F}_T^1=C_T\mathcal{F}_T$, and hence the metric and kernel can be reconstructed from the response function $r(t)=\sum_k\rho_k^{-1}S_k(t)$.
The spectral data $\{\lambda_k,\rho_k\}$ determine $B_N$ as the span of the first $N$ orthogonal polynomials with the $L^2_\rho$ metric, so the de Branges structure is a repackaging of the classical orthogonal-polynomial spectral theory of the Jacobi matrix.
Applying the criterion of Theorem 3 to the reproducing kernel produces an explicit Hermite-Biehler function $E$ for the finite Jacobi system, placing the system inside the de Branges-space framework used for canonical systems.
The construction works despite the two features that blocked earlier applications: the system lacks boundary controllability on the whole state space, and the wave propagation speed is infinite.

Reading between the lines

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

The same template should extend to any second-order boundary-control system admitting an analogue of Lemma 2: a finite-dimensional family of controls that maps isomorphically onto the states and whose connecting operator is accessible from inverse data. This generality is the authors' implicit suggestion, not a claim of the paper.
Taking $N\to\infty$ would plausibly produce de Branges spaces of entire functions associated with infinite Jacobi matrices as limits of $B_N$; the paper mentions the semi-infinite discrete case only through a citation and does not develop this limit.
Because $B_N=L_N$ as a metric space, the reproducing kernel $J_z$ is the Christoffel-Darboux kernel of the Jacobi polynomials $\phi_k$; a numerical comparison of this kernel with the one produced by the Krein equations for small $N$ would independently test the construction.

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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

The construction works, but the proof has a real error in the de Branges verification. The reader only caught the conjugation slip in the reproducing kernel; the stress-test is right that something more serious is there: condition 3 of Theorem 3, as invoked, uses (z−ω)/(z−ω̄), which has a pole at z = ω̄ for non-real ω. The function then cannot be entire, so the verification as written is invalid. The correct multiplier is (z−ω̄)/(z−ω), which cancels the zero at ω and preserves polynomial degree. In this setting the corrected condition holds, since |λ−ω̄| = |λ−ω| on R, so the main claim is repairable. The paper should state the theorem correctly and run the verification with the reciprocal multiplier.

What is genuinely new here is the application of the authors' boundary-control construction of de Branges spaces to the second-order wave equation associated with a finite Jacobi matrix. Unlike the Schrödinger and Dirac cases treated earlier, this system has infinite propagation speed and lacks exact controllability; the paper shows the method still produces a de Branges space. The resulting space is the standard polynomial space with the spectral measure, so the outcome is not surprising, but the demonstration is legitimate. The norm on B_N is defined through the connecting operator, which is expressible in terms of inverse data, so there is no circularity. Lemma 2 is imported from prior work, and the proof is terse, but the essential argument is there.

The conjugation error in (J_z, G) = G(z) is real but minor: with the Hermitian inner product on H_N, the prescribed state must be $\overline{\phi_k(z)}$, not φ_k(z), for the reproducing identity to hold. This does not affect the de Branges property. Heavy self-citation is present, but the cited lemmas are substantive.

I would send this to a referee. The main theorem is correct and the defects are local and fixable. The paper deserves a serious review, and I expect it to be accepted after a careful revision. My advice: engage with it, but insist the authors correct Theorem 3's condition and the kernel identity before publication.

Referee Report

2 major / 4 minor

Summary. The paper considers a boundary control dynamical system (utt − Au = F, with finite Jacobi matrix A) and, following the authors' earlier boundary control method, constructs a finite-dimensional space B_N of polynomials (degree at most N−1) as the Fourier image of the reachable set. The space is equipped with an inner product defined through the connecting operator C_T, a candidate reproducing kernel J_z is introduced via specially chosen controls, and the paper claims that B_N is a de Branges space by verifying the three conditions of Theorem 3. The novelty is that this system has infinite propagation speed and is not exactly controllable, in contrast to earlier models, yet the de Branges space construction is asserted to go through.

Significance. If the construction is valid, the paper extends the boundary-control approach to de Branges spaces to a new class of systems with infinite wave speed and without exact controllability, complementing the authors' previous results for Schrödinger, Dirac, and discrete Jacobi systems. The finite-dimensional model is explicit and the reliance on Lemma 2 (special boundary controllability) is a clear and testable mechanism. However, two technical flaws in the current proof must be corrected before the central claim can be accepted.

major comments (2)

[Section 3, reproducing kernel] Condition 3 of Theorem 3 is misstated. The displayed multiplier is (z−ω)/(z−ω̄), but for non-real ω this expression is not entire unless f also vanishes at ω̄. The correct de Branges axiom uses the reciprocal multiplier (z−ω̄)/(z−ω). The verification in Section 3 repeats this error: for N≥2, take G(z)=z−i and ω=i; then ((z−i)/(z+i))G(z) = (z−i)^2/(z+i), which has a pole at z=−i and hence cannot belong to a space of entire functions. Thus the proof of 'Thus B_N is a de Branges space' is invalid as written. The claim is nonetheless repairable: since G(ω)=0, one can write G(z)=(z−ω)Q(z) with deg Q ≤ N−2, and the corrected multiplier gives (z−ω̄)Q(z), which is a polynomial of degree at most N−1, and the norm equality follows from |(λ−ω̄)/(λ−ω)|=1 for real λ. The authors must correct both the statement of Theorem 3 and the verification.
The identity (J_z, G)_{B_N} = G(z) is not correct for the Hermitian inner product defined by (H,G)_{B_T} = ∫ H(λ) \overline{G(λ)} dρ(λ). From the definitions one obtains (J_z, G)_{B_N} = ∑_{k=1}^N ϕ_k(z) \overline{c_k} if G=∑ c_k ϕ_k, which equals \overline{G(\bar z)} (because the ϕ_k have real coefficients), not G(z). Consequently J_z as defined is not the reproducing kernel for this inner product; the correct kernel is K_z(λ)=∑ \overline{ϕ_k(z)} ϕ_k(λ). This error affects the proof of condition 1 of Theorem 3, where the pointwise bound is derived from the asserted kernel identity. The bound itself can still be proven, for example by using K_{\bar z} or directly from the basis expansion, so this is a fixable defect, but the argument as written is incorrect.

minor comments (4)

[Section 3, norm computation] In the chain of equalities computing (H,G)_{B_T}, the integrand writes \overline{(F u_h(T))(λ)} where the second factor should be (F u_g(T))(λ); the final expression H(λ)\overline{G(λ)} is correct but the intermediate display is a typo.
[Section 3, verification of condition 3] The text uses the symbols F and G inconsistently: 'When ω∈C such that F(ω)=0' is followed by 'z−ω/z−ω̄ F(z)' and then 'z−ω/z−ω̄ G(z)' and finally '=||G||_BN'. This should be cleaned up.
[Introduction] Reference [9] is cited twice as '[9, 9]', and the abstract contains the typo 'de Banges spaces'.
[Section 3, definition of B(E)] For completeness, the definition of the de Branges space B(E) should specify that E is a Hermite–Biehler function and that the quotient conditions are required to hold in the Hardy space; the current presentation is terse but acceptable.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the de Branges construction is self-contained apart from prior controllability lemmas cited from the authors' own work.

full rationale

The paper's derivation chain is: Lemma 2 (quoted from [7,8]) gives an isomorphism W_T: F_T^1 -> H_N, so the reachable set is spanned by spectral data; the inner product on B_N = F U_T is defined through the connecting operator C_T, which is independently expressed in Theorem 1 in terms of the response function and spectral data; and the de Branges property is then verified against the external characterization Theorem 3, not assumed. The reproducing kernel identity (J_z, G) = G(z) follows from the definition of the special control j_z and from the definition of the metric, so it is a derived property, not an input. No parameter is fitted to a subset of data and then renamed a prediction; the connecting operator is not defined in terms of the de Branges norm, but the other way around. The heavy self-citation is to earlier controllability and BC-method results that are independent of the target claim and whose assumptions do not include the de Branges property; thus under the stated rules those citations count as external support. A possible misstatement of the multiplier in condition 3 of Theorem 3 is a mathematical correctness concern, not a circularity, and does not raise the circularity score.

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

No fitted constants or invented entities. The arguments rest on standard orthogonal polynomial theory, de Branges space theory, and the authors' prior controllability and boundary control results, which are cited but not re-derived.

assumptions (4)

standard math Finite Jacobi matrix spectral theorem and discrete orthogonality of the polynomials φ_k in L^2_ρ
Used in Section 2 (Lemma 1) for the spectral representation and in Section 3 to establish the unitarity of the Fourier transform and the equality (H,G)_{B_T} = ∫ H \bar{G} dρ.
domain assumption Boundary controllability of F_T^1 (Lemma 2) from Mikhaylov & Mikhaylov [7,8]
Identifies the reachable set U_T = W_T F_T with W_T F_T^1, ensuring the metric on B_N is well-defined and nondegenerate.
domain assumption Connecting operator representation and Krein equations (Theorems 1 and 2) from [7,8]
Expresses C_T in terms of inverse data and guarantees existence and uniqueness of the special control j_z used to build the reproducing kernel.
standard math De Branges converse characterization (Theorem 3), cited from [5,6]
Provides the three axioms used to verify that B_N is a de Branges space.

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Cite this review

Pith. "Pith review of On the construction of de Branges spaces for dynamical systems associated with finite Jacobi matrices." pith.science (2026). https://pith.science/paper/AOBODQW5

@misc{pith2026250520905,
  author       = {Pith},
  title        = {Pith review of: On the construction of de Branges spaces for dynamical systems associated with finite Jacobi matrices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AOBODQW5}},
  note         = {Machine review of arXiv:2505.20905}
}

read the original abstract

We consider dynamical systems with boundary control associated with finite Jacobi matrices. Using the method previously developed by the authors, we associate with these systems special Hilbert spaces of analytic functions (de Branges spaces)

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Reference graph

Works this paper leans on

15 extracted references · 13 canonical work pages

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[9]

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A. S. Mikhaylov, V. S. Mikhaylov. Inverse problem for dynamical system associated with Jacobi matrices and classical moment problems. Journal of Mathematical Analysis and Applications, 487, no. 1, 2020

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Akhiezer

N.I. Akhiezer . The classical moment problem. Oliver and Boyd, Edinburgh. 1965

work page 1965

[2] [2]

M. I. Belishev. Boundary control in reconstruction of manifolds and metrics (the BC method). Inverse Problems, 13, no 5, R1--R45, 1997

work page 1997

[3] [3]

M. I. Belishev. Boundary control and inverse problems: a one-dimensional version of the boundary control method. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 354, 2008, 19-80 (in Russian); English translation: J. Math. Sci. (N. Y.), 155, no 3, 2008, 343--378

work page 2008

[4] [4]

I.Belishev

M. I.Belishev . Recent progress in the boundary control method. Inverse Problems, 23, no 5, 2007, R1--R67

work page 2007

[5] [5]

Hilbert space of entire functions

Louis de Branges . Hilbert space of entire functions. Prentice-Hall, NJ. 1968

1968

[6] [6]

H Dym, H. P. McKean . Gaussian processes, function theory, and the inverse spectral problem. Academic Press, New York etc. 1976

work page 1976

[7] [7]

A. S. Mikhaylov, V. S. Mikhaylov. Dynamic inverse problem for a Krein--Stieltjes string. Applied Mathematics Letters, 96, 195--210, 2019

work page 2019

[8] [8]

A. S. Mikhaylov, V. S. Mikhaylov. Inverse dynamic problem for finite Jacobi matrices. Journal of Inverse and Ill-Posed problems, 29, no. 4, 611-628, 2021

work page 2021