{
  "id": "2406.19916",
  "version": "v1",
  "published": "2024-06-28T13:39:59.000Z",
  "updated": "2024-06-28T13:39:59.000Z",
  "title": "The truncated multidimensional moment problem: canonical solutions",
  "authors": [
    "Sergey M. Zagorodnyuk"
  ],
  "comment": "25 pages",
  "categories": [
    "math.CA"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-28T13:39:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "44A60"
    ],
    "keywords": [
      "truncated multidimensional moment problem",
      "canonical solution",
      "conditions",
      "two-dimensional moment problem",
      "commuting self-adjoint extensions inside"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}