The Multidimensional Moment Problem with Complexity Constraint

TL;DR

This paper provides a comprehensive solution to the multidimensional moment problem with a complexity constraint, including solutions with singular measures, expanding the understanding of the solution space.

Contribution

It offers a complete parameterization of all solutions, including those with singular measures, for the multidimensional moment problem under complexity constraints.

Findings

Complete solution for multidimensional moment problem with complexity constraint.

Parameterization of solutions including singular measures.

Addresses boundary solutions on a convex cone.

Abstract

A long series of previous papers have been devoted to the (one-dimensional) moment problem with nonnegative rational measure. The rationality assumption is a complexity constraint motivated by applications where a parameterization of the solution set in terms of a bounded finite number of parameters is required. In this paper we provide a complete solution of the multidimensional moment problem with a complexity constraint also allowing for solutions that require a singular measure added to the rational, absolutely continuous one. Such solutions occur on the boundary of a certain convex cone of solutions. In this paper we provide complete parameterizations of all such solutions. We also provide errata for a previous paper in this journal coauthored by one of the authors of the present paper.