{
  "id": "2109.06833",
  "version": "v1",
  "published": "2021-09-14T17:17:34.000Z",
  "updated": "2021-09-14T17:17:34.000Z",
  "title": "On the best Ulam constant of the linear differential operator with constant coefficients",
  "authors": [
    "Alina-Ramona Baias",
    "Dorian Popa"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "The linear differential operator with constant coefficients $$D(y)=y^{(n)}+a_1 y^{(n-1)}+\\ldots+a_n y,\\quad y\\in \\mathcal{C}^{n}(\\mathbb{R}, X)$$ acting in a Banach space $X$ is Ulam stable if and only if its characteristic equation has no roots on the imaginary axis. We prove that if the characteristic equation of $D$ has distinct roots $r_k$ satisfying $\\Real r_k>0,$ $1\\leq k\\le n,$ then the best Ulam constant of $D$ is $K_D=\\frac{1}{|V|}\\int_{0}^{\\infty}\\left|\\sum\\limits_{k=1}^n(-1)^kV_ke^{-r_k x}\\right|dx,$ where $V=V(r_1,r_2,\\ldots,r_n)$ and $V_k=V(r_1,\\ldots,r_{k-1},r_{k+1}, \\ldots, r_n),$ $1\\leq k\\leq n,$ are Vandermonde determinants.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-14T17:17:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "linear differential operator",
      "best ulam constant",
      "constant coefficients",
      "characteristic equation",
      "banach space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}