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arxiv: 1907.11153 · v1 · pith:6MJGYAUGnew · submitted 2019-07-25 · 🧮 math.AP · math.DS· math.SP

Dynamic inverse problem for special system associated with Jacobi matrices and classical moment problems

Alexander Mikhaylov , Victor Mikhaylov This is my paper

Pith reviewed 2026-05-24 15:55 UTC · model grok-4.3

classification 🧮 math.AP math.DSmath.SP
keywords moment problemsJacobi matrixinverse dynamic problemHankel matricesspectral measureboundary controlHamburger moment problemtruncated moment problem
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The pith

The moments of a measure determine the input-output map of an auxiliary discrete dynamical system linked to a Jacobi matrix, allowing recovery of truncated spectral measures via finite generalized eigenvalue problems.

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

The paper links the classical Hamburger, Stieltjes and Hausdorff moment problems to an auxiliary discrete-time dynamical system built from a semi-infinite Jacobi matrix. It shows that any given moment sequence directly supplies the inverse dynamic data of this system. Boundary-control reconstruction then recovers the spectral measure supported on the N by N principal submatrix for each finite N by solving a generalized eigenvalue problem whose coefficient matrices are assembled from the moments through explicit formulas involving Hankel matrices. Existence and uniqueness statements for the three moment problems are thereby expressed in terms of the same matrices and Krein-type inverse equations.

Core claim

The set of moments determines the inverse dynamic data for the auxiliary system associated with the semi-infinite Jacobi matrix; for every N the spectral measure of the N by N block is recovered by solving a finite-dimensional generalized spectral problem whose matrices are constructed from the moments via simple formulas involving Hankel matrices.

What carries the argument

Auxiliary discrete-time dynamical system associated with the semi-infinite Jacobi matrix, whose input-output map is encoded exactly by the moment sequence.

If this is right

  • For each N the spectral measure of the N by N Jacobi block solves the corresponding truncated moment problem.
  • Existence of a solution to any of the three moment problems is equivalent to the constructed Hankel-related matrices admitting a positive solution to the finite generalized spectral problem.
  • Uniqueness of the measure follows from the Krein-type equations once the dynamic data are recovered from the moments.
  • The same construction supplies a uniform numerical procedure for the truncated versions of all three classical moment problems.

Where Pith is reading between the lines

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

  • The finite generalized spectral problems could be used to generate stable numerical schemes that avoid explicit orthogonal-polynomial computation.
  • The dynamic-system viewpoint may extend to moment problems on more general graphs or to inverse problems for non-self-adjoint Jacobi operators.
  • Because the matrices are built directly from Hankel forms, the method supplies an explicit link between the classical moment conditions and the positive-definiteness criteria used in the boundary-control approach.

Load-bearing premise

The auxiliary discrete-time dynamical system can be associated with the semi-infinite Jacobi matrix so that the given moment sequence exactly encodes its input-output map.

What would settle it

Compute the matrices from the moments of the uniform measure on [0,1], solve the generalized eigenvalue problem for N=3, and check whether the resulting three-point measure reproduces the first six original moments.

read the original abstract

We consider the Hamburger, Stieltjes and Hausdorff moment problems, that are problems of the construction of a Borel measure supported on a real line, on a half-line or on an interval $(0,1)$, from a prescribed set of moments. We propose a unified approach to these three problems based on using the auxiliary dynamical system with the discrete time associated with a semi-infinite Jacobi matrix. It is show that the set of moments determines the inverse dynamic data for such a system. Using the ideas of the Boundary Control method for every $N\in \mathbb{N}$ we can recover the spectral measure of a $N\times N$ block of Jacobi matrix, which is a solution to a truncated moment problem. This problem is reduced to the finite-dimensional generalized spectral problem, whose matrices are constructed from moments and are connected with well-known Hankel matrices by simple formulas. Thus the results on existence of solutions to Hamburger, Stieltjes and Hausdorff moment problems are naturally given in terms of these matrices. We also obtain results on uniqueness of the solution of moment problems, where as a main tool we use the Krein-type equations of inverse problem.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript proposes a unified approach to the Hamburger, Stieltjes, and Hausdorff moment problems by associating each with an auxiliary discrete-time dynamical system linked to a semi-infinite Jacobi matrix. It asserts that the prescribed moment sequence determines the inverse dynamic data (input-output map) of this system, and that for each finite N the spectral measure of the N×N principal block—which solves the corresponding truncated moment problem—is recovered by solving a finite-dimensional generalized eigenvalue problem whose two matrices are constructed explicitly from the moments via simple formulas involving the associated Hankel matrices. Existence and uniqueness statements for the three moment problems are then obtained in terms of these matrices, with Krein-type equations of the inverse problem used as the main tool for uniqueness.

Significance. If the central derivations hold, the work supplies a constructive, dynamical-systems route to the classical moment problems that reduces each truncated problem to an explicit generalized spectral problem built directly from Hankel matrices. This yields both existence criteria and uniqueness results phrased in terms of the same matrices, together with a link to boundary-control reconstruction techniques. The explicit, parameter-free character of the matrix constructions from the moments is a notable strength that could support both theoretical comparisons and numerical implementations.

minor comments (3)
  1. Abstract, line 3: 'It is show that' should read 'It is shown that'.
  2. The manuscript would benefit from an explicit statement, early in the introduction or §2, of the precise correspondence between the moment sequence and the input-output map of the auxiliary system (i.e., which moments correspond to which entries of the dynamic response operator).
  3. Notation for the finite sections of the Jacobi matrix and the associated Hankel matrices should be introduced once and used consistently; at present the same symbols appear to be reused for both the infinite and truncated objects in several places.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive assessment of its contributions. The recommendation for minor revision is noted; however, the report contains no enumerated major comments requiring specific responses.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via explicit Hankel constructions

full rationale

The paper constructs matrices for the generalized eigenvalue problem directly from the moment sequence using explicit formulas tied to Hankel matrices, then solves the truncated moment problem for each finite N. This is a direct algebraic reduction rather than a fitted parameter renamed as a prediction or a self-referential definition. The auxiliary dynamical system is introduced as the unifying framework whose input-output map is shown to be encoded by the moments; the Boundary Control recovery step then operates on that map without reducing back to the original moments by construction. No load-bearing self-citation chain or imported uniqueness theorem is required for the central claim, and the results on existence/uniqueness are stated in terms of the same explicitly constructed matrices. The derivation therefore remains independent of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on the standard theory of Jacobi matrices and the Boundary Control method for discrete systems; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption A semi-infinite Jacobi matrix generates a discrete-time dynamical system whose input-output map is determined by the moment sequence of the associated spectral measure.
    Invoked at the outset to equate the moment problem with an inverse dynamic problem.
  • domain assumption The Boundary Control method applies to the finite initial blocks of this system and yields the spectral measure of each block.
    Central to the reconstruction step for every N.

pith-pipeline@v0.9.0 · 5735 in / 1501 out tokens · 18703 ms · 2026-05-24T15:55:57.762248+00:00 · methodology

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