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Multidimensional moment problem and diagonal Schur algorithm

Ivan Kovalyov

TL;DR

This work develops a diagonal Schur-type framework for the multidimensional Stieltjes moment problem by reformulating it through the $n$-variate Stieltjes transform and exploiting diagonal reductions. By isolating diagonal moment structures ($d_0$, then $d_{j_1,...,j_n}$) and associated moment sequences, the authors derive explicit continued-fraction (S-fraction) representations and Stieltjes-polynomial descriptions for all solutions, including truncated and full problems. They introduce diagonal atoms and resolvent matrices, provide regularity conditions, and obtain determinacy criteria. The approach unifies the multidimensional problem with well-understood one-dimensional techniques, yielding constructive solutions and a pathway to analyze convergence and indeterminacy across the full problem through its diagonal components.

Abstract

The multidimensional moment problem is studied in terms of the Steiltjes transform. The diagonal step-by-step algorithm is constructed for the multidimensional moment problem. The set of solutions of the full multidimensional moment problem is found in terms of the continued fractions. Moreover, the diagonal step-by-step algorithm can be applied to the special truncated multidimensional moment problem.

Multidimensional moment problem and diagonal Schur algorithm

TL;DR

This work develops a diagonal Schur-type framework for the multidimensional Stieltjes moment problem by reformulating it through the -variate Stieltjes transform and exploiting diagonal reductions. By isolating diagonal moment structures (, then ) and associated moment sequences, the authors derive explicit continued-fraction (S-fraction) representations and Stieltjes-polynomial descriptions for all solutions, including truncated and full problems. They introduce diagonal atoms and resolvent matrices, provide regularity conditions, and obtain determinacy criteria. The approach unifies the multidimensional problem with well-understood one-dimensional techniques, yielding constructive solutions and a pathway to analyze convergence and indeterminacy across the full problem through its diagonal components.

Abstract

The multidimensional moment problem is studied in terms of the Steiltjes transform. The diagonal step-by-step algorithm is constructed for the multidimensional moment problem. The set of solutions of the full multidimensional moment problem is found in terms of the continued fractions. Moreover, the diagonal step-by-step algorithm can be applied to the special truncated multidimensional moment problem.
Paper Structure (9 sections, 19 theorems, 68 equations)

This paper contains 9 sections, 19 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Multidimensional moment problem
  3. Diagonal type
  4. $d_0$-diagonal full problem
  5. $d_{j_1,...,j_n}$-diagonal types
  6. Odd $d_{j_1,...,j_n}$-diagonal truncated problem
  7. Even $d_{j_1,...,j_n}$-diagonal truncated problem
  8. $d_{j_1,...,j_n}$-diagonal full problem
  9. Full problem in general case

Key Result

Theorem 1.1

(Der03) Let $\mathbf{s}=\{s_{j}\}_{j=0}^{\ell}$ be a sequence of real numbers and assume that $\ell>2n_1-1$, where $n_1$ is the first normal index of the sequence $\mathbf{s}$. If $f$ admits the asymptotic expansion 14.int.th_1, then $f$ cam be rewritten as where the recursive sequence $\mathbf{s}^{(1)}=\left\{s_j^{(1)}\right\}_{j=0}^{\ell-2}$ is defined by In DK15DK17K17, $MP(\mathbf{s},\ell)$

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Remark 1.5
  • Proposition 2.1
  • Proposition 2.2
  • Remark 2.3
  • Definition 2.4
  • Proposition 2.5
  • ...and 19 more