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On the complex moment problem as a dynamic inverse problem for a discrete system

A. S. Mikhaylov, V. S. Mikhaylov

TL;DR

This work addresses the complex moment problem by framing it as a dynamic inverse problem for a discrete system governed by a complex Jacobi matrix. It develops a boundary-control approach and a spectral representation based on Autonne–Takagi factorization to connect moment data with a discrete spectral measure. A constructive procedure is provided to solve the truncated complex moment problem and, under non-singularity conditions, to obtain a limiting measure solving the full problem; the method yields discrete measures with explicit support and masses linked to the Jacobi data and the dynamic inverse data. The approach offers a practical pathway from boundary-control-derived inverse data to measures solving the complex moment problem, without requiring boundedness of moments, and paves the way for further connections with generalized spectral functions.

Abstract

We consider the complex moment problem, that is the problem of constructing a positive Borel measure on $\mathbb{C}$ from a given set of moments. We relate this problem to the dynamic inverse problem for the discrete system associated with the complex Jacobi matrix. We show how the characterization of dynamic inverse data in solving the inverse problem provides sufficient conditions for solving the complex moment problem.

On the complex moment problem as a dynamic inverse problem for a discrete system

TL;DR

This work addresses the complex moment problem by framing it as a dynamic inverse problem for a discrete system governed by a complex Jacobi matrix. It develops a boundary-control approach and a spectral representation based on Autonne–Takagi factorization to connect moment data with a discrete spectral measure. A constructive procedure is provided to solve the truncated complex moment problem and, under non-singularity conditions, to obtain a limiting measure solving the full problem; the method yields discrete measures with explicit support and masses linked to the Jacobi data and the dynamic inverse data. The approach offers a practical pathway from boundary-control-derived inverse data to measures solving the complex moment problem, without requiring boundedness of moments, and paves the way for further connections with generalized spectral functions.

Abstract

We consider the complex moment problem, that is the problem of constructing a positive Borel measure on from a given set of moments. We relate this problem to the dynamic inverse problem for the discrete system associated with the complex Jacobi matrix. We show how the characterization of dynamic inverse data in solving the inverse problem provides sufficient conditions for solving the complex moment problem.

Paper Structure

This paper contains 5 sections, 10 theorems, 57 equations.

Table of Contents

  1. Introduction.
  2. Discrete dynamical system. Forward problem, special representation of the solution.
  3. Inverse problem for discrete dynamical system associated with complex Jacobi matrices.
  4. Inverse problem.
  5. Complex moment problem and dynamic inverse problem.

Key Result

Theorem 1

Let $H\in \mathbb{C}^{n\times n}$ be a complex symmetric matrix: $H^*=\overline H$, then there exists a unitary matrix $U$ such that where $\hat{d}_i\geqslant 0,$$i=1\ldots,n$.

Theorems & Definitions (15)

  • Theorem 1
  • Lemma 1
  • Definition 1
  • Definition 2
  • Lemma 2
  • Lemma 3
  • Theorem 2
  • Remark 1
  • Remark 2
  • Theorem 3
  • ...and 5 more