The truncated multidimensional moment problem: canonical solutions
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910506476896256 |
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| author | Zagorodnyuk, Sergey M. |
| author_facet | Zagorodnyuk, Sergey M. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_19916 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The truncated multidimensional moment problem: canonical solutions Zagorodnyuk, Sergey M. Classical Analysis and ODEs 44A60 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. |
| title | The truncated multidimensional moment problem: canonical solutions |
| topic | Classical Analysis and ODEs 44A60 |
| url | https://arxiv.org/abs/2406.19916 |