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arXiv · 2406.19916

The truncated multidimensional moment problem: canonical solutions

Sergey M. Zagorodnyuk
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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.

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Sergey M. Zagorodnyuk. 2024-06-28. The truncated multidimensional moment problem: canonical solutions. https://arxiv.org/abs/2406.19916

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