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Two-dimensional moment problem and Schur algorithm

Ivan Kovalyov, Stefan Kunis

TL;DR

This work develops a two-dimensional Stieltjes moment framework using a Schur algorithm-based, multi-variable continued-fraction approach to characterize the solution set of truncated and full 2D moment problems. By formulating the problem through the two-variable transform $F(z,\zeta)$ and introducing simple regular and simple plus regular sequences, the authors derive explicit continued-fraction representations, along with atom-based constructions $(a_j,b_j)$ (and $(m_i,l_i)$ in the Stieltjes-like setting) and corresponding recursive schemes. The treatment covers non-symmetric and symmetric forms, basic and general truncations, nth convergents, and an atomic-measure specialization, culminating in a convergent framework and reconstruction strategies from finite or infinite moment data. The results extend to atomic measures and provide practical tools for solving higher-dimensional moment problems via Schur-type iterations and linear-fractional transformations, with explicit convergence criteria.

Abstract

We study a truncated two-dimensional moment problem in terms of the Stieltjes transform. The set of the solutions is described by the Schur step-by-step algorithm, which is based on the continued fraction expansion of the solution. In particular, the obtained results are applicable to the two-dimensional moment problem for atomic measures.

Two-dimensional moment problem and Schur algorithm

TL;DR

This work develops a two-dimensional Stieltjes moment framework using a Schur algorithm-based, multi-variable continued-fraction approach to characterize the solution set of truncated and full 2D moment problems. By formulating the problem through the two-variable transform and introducing simple regular and simple plus regular sequences, the authors derive explicit continued-fraction representations, along with atom-based constructions (and in the Stieltjes-like setting) and corresponding recursive schemes. The treatment covers non-symmetric and symmetric forms, basic and general truncations, nth convergents, and an atomic-measure specialization, culminating in a convergent framework and reconstruction strategies from finite or infinite moment data. The results extend to atomic measures and provide practical tools for solving higher-dimensional moment problems via Schur-type iterations and linear-fractional transformations, with explicit convergence criteria.

Abstract

We study a truncated two-dimensional moment problem in terms of the Stieltjes transform. The set of the solutions is described by the Schur step-by-step algorithm, which is based on the continued fraction expansion of the solution. In particular, the obtained results are applicable to the two-dimensional moment problem for atomic measures.
Paper Structure (15 sections, 13 theorems, 115 equations)

This paper contains 15 sections, 13 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Non-symmetric case
  3. Basic truncated problem
  4. Truncated problem $MP(\mathbf{s},\mathfrak{N}(\mathbf{s}))$
  5. Symmetric form
  6. Basic 1st type
  7. Basic 2nd type
  8. General case
  9. The $n$-th convergent
  10. Stieltjes like case
  11. Basic problem
  12. General case
  13. The $n$-th convergent
  14. Moment problem for atomic measures
  15. Full problem

Key Result

Theorem 1.1

(Akh) Let $\mathbf{s}=\{s_{j}\}_{j=0}^{\ell}$ be a sequence of real numbers with $s_0\neq 0$ and let $f$ admit the asymptotic expansion 14.int.th_1 Then $f$ takes the form where the recursive sequence $\mathbf{s}^{(1)}=\left\{s_j^{(1)}\right\}_{j=0}^{\ell-2}$ is defined by

Theorems & Definitions (29)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 19 more