{
  "id": "2011.14987",
  "version": "v1",
  "published": "2020-11-30T16:57:04.000Z",
  "updated": "2020-11-30T16:57:04.000Z",
  "title": "Universal relations in asymptotic formulas for orthogonal polynomials",
  "authors": [
    "D. R. Yafaev"
  ],
  "categories": [
    "math.CA",
    "math.FA",
    "math.SP"
  ],
  "abstract": "Orthogonal polynomials $P_{n}(\\lambda)$ are oscillating functions of $n$ as $n\\to\\infty$ for $\\lambda$ in the absolutely continuous spectrum of the corresponding Jacobi operator $J$. We show that, irrespective of any specific assumptions on coefficients of the operator $J$, amplitude and phase factors in asymptotic formulas for $P_{n}(\\lambda)$ are linked by certain universal relations found in the paper. Our approach relies on a study of operators diagonalizing Jacobi operators. Diagonalizing operators are constructed in terms of orthogonal polynomials $P_{n}(\\lambda)$. They act from the space $L^2 (\\Bbb R)$ of functions into the space $\\ell^2 ({\\Bbb Z}_{+})$ of sequences. We consider such operators in a rather general setting and find necessary and sufficient conditions of their boundedness.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-30T16:57:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C45",
      "47B38"
    ],
    "keywords": [
      "orthogonal polynomials",
      "asymptotic formulas",
      "universal relations",
      "operators diagonalizing jacobi operators",
      "approach relies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}