{
  "id": "math-ph/0605011",
  "version": "v1",
  "published": "2006-05-04T09:11:35.000Z",
  "updated": "2006-05-04T09:11:35.000Z",
  "title": "Generalization of a theorem of Carathéodory",
  "authors": [
    "Salvino Ciccariello",
    "Antonio Cervellino"
  ],
  "comment": "30 pages; submitted to J. Phys. A - Math. Gen",
  "doi": "10.1088/0305-4470/39/48/006",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "Carath\\'eodory showed that $n$ complex numbers $c_1,...,c_n$ can uniquely be written in the form $c_p=\\sum_{j=1}^m \\rho_j {\\epsilon_j}^p$ with $p=1,...,n$, where the $\\epsilon_j$s are different unimodular complex numbers, the $\\rho_j$s are strictly positive numbers and integer $m$ never exceeds $n$. We give the conditions to be obeyed for the former property to hold true if the $\\rho_j$s are simply required to be real and different from zero. It turns out that the number of the possible choices of the signs of the $\\rho_j$s are {at most} equal to the number of the different eigenvalues of the Hermitian Toeplitz matrix whose $i,j$-th entry is $c_{j-i}$, where $c_{-p}$ is equal to the complex conjugate of $c_{p}$ and $c_{0}=0$. This generalization is relevant for neutron scattering. Its proof is made possible by a lemma - which is an interesting side result - that establishes a necessary and sufficient condition for the unimodularity of the roots of a polynomial based only on the polynomial coefficients. Keywords: Toeplitz matrix factorization, unimodular roots, neutron scattering, signal theory, inverse problems. PACS: 61.12.Bt, 02.30.Zz, 89.70.+c, 02.10.Yn, 02.50.Ga",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-05-04T09:11:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11L03",
      "30C15",
      "15A23",
      "15A90",
      "42A63",
      "42A70",
      "42A82"
    ],
    "keywords": [
      "generalization",
      "carathéodory",
      "unimodular complex numbers",
      "hermitian toeplitz matrix",
      "toeplitz matrix factorization"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}