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Multidimensional moment problem and Stieltjes transform

Ivan Kovalyov

TL;DR

The paper tackles the truncated multidimensional moment problem using the Stieltjes transform as an interpolation problem, and develops a constructive, step-by-step algorithm to characterize all solutions. It shows that the problem can be decomposed into a family of one-dimensional problems along slices indexed by the remaining variables, enabling explicit representations via P-fractions and S-fractions. The truncated problem yields finite-sum representations across the slices with atoms derived from moments, while the full problem is treated via infinite continued fractions and indeterminacy criteria tied to the convergence of tail terms. The approach unifies multidimensional moment problems with classical one-dimensional continued-fraction theory and provides practical recipes for computing solution families through Stieltjes polynomials and associated shift techniques. This framework advances both the theoretical understanding and the computational handling of multidimensional moment interpolation problems using Schur-type algorithms.

Abstract

The truncated multidimensional moment problem is studied in terms of the Stieltjes transform as the interpolation problem. A step-by-step algorithm is constructed for the multidimensional moment problem and the set of solutions is found in terms of continued fractions.

Multidimensional moment problem and Stieltjes transform

TL;DR

The paper tackles the truncated multidimensional moment problem using the Stieltjes transform as an interpolation problem, and develops a constructive, step-by-step algorithm to characterize all solutions. It shows that the problem can be decomposed into a family of one-dimensional problems along slices indexed by the remaining variables, enabling explicit representations via P-fractions and S-fractions. The truncated problem yields finite-sum representations across the slices with atoms derived from moments, while the full problem is treated via infinite continued fractions and indeterminacy criteria tied to the convergence of tail terms. The approach unifies multidimensional moment problems with classical one-dimensional continued-fraction theory and provides practical recipes for computing solution families through Stieltjes polynomials and associated shift techniques. This framework advances both the theoretical understanding and the computational handling of multidimensional moment interpolation problems using Schur-type algorithms.

Abstract

The truncated multidimensional moment problem is studied in terms of the Stieltjes transform as the interpolation problem. A step-by-step algorithm is constructed for the multidimensional moment problem and the set of solutions is found in terms of continued fractions.
Paper Structure (7 sections, 12 theorems, 68 equations)

This paper contains 7 sections, 12 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Indefinite one-dimensional moment problem
  3. Truncated problem and P-fractions
  4. Truncated problem and S-fractions
  5. Regular case
  6. $\alpha$-regular case
  7. Full problem

Key Result

Theorem 2.1

(Der03) Let $\mathbf{s}=\{s_{j}\}_{j=0}^{2n_N-1}$ be a sequence of real numbers and let ${\mathcal{N}}({\mathbf s})=\{n_j\}_{j=1}^N$ be the set of normal indices of ${\mathbf s}$. Then any solution of the moment problem $MP(\mathbf{s}, 2n_N-1)$ takes the form where the prameter $\tau$ satisfies the following condition and atoms $(a_i,b_i)$ can be calculated by where $i=\overline{0,2\nu-1}$, $j=

Theorems & Definitions (18)

  • Definition 1.1
  • Theorem 2.1
  • Remark 2.2
  • Definition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 3.1
  • Corollary 3.2
  • Theorem 4.1
  • Corollary 4.2
  • ...and 8 more