The truncated multidimensional moment problem: canonical solutions

TL;DR

This paper introduces the concept of canonical solutions for the truncated multidimensional moment problem, establishing a correspondence with flat extensions and providing explicit conditions for their existence in various cases.

Contribution

It defines canonical solutions generated by commuting self-adjoint extensions and characterizes their existence through algebraic and quadratic conditions, including the index of nonself-adjointness.

Findings

Established a 1-1 correspondence between canonical solutions and flat extensions.

Derived explicit necessary and sufficient conditions for canonical solutions when index is 1.

Provided numerical examples illustrating the theoretical results.

Abstract

For the truncated multidimensional moment problem we introduce a notion of a canonical solution. Namely, canonical solutions are those solutions which are generated by commuting self-adjoint extensions inside the associated Hilbert space. It is constructed a 1-1 correspondence between canonical solutions and flat extensions of the given moments (both sets may be empty). In the case of the two-dimensional moment problem (with triangular truncations) a search for canonical solutions leads to an algebraic system of equations. A notion of the index $i_s$ of nonself-adjointness for a set of prescribed moments is introduced. The case $i_s=0$ corresponds to flatness. In the case $i_s=1$ we get explicit necessary and sufficient conditions for the existence of canonical solutions. These conditions are valid for arbitrary sizes of truncations. In the case $i_s=2$ we get either explicit conditions for the existence of canonical solutions or a single quadratic equation with several unknowns. Numerical examples are provided.