{
  "id": "2401.11245",
  "version": "v1",
  "published": "2024-01-20T14:57:15.000Z",
  "updated": "2024-01-20T14:57:15.000Z",
  "title": "Construction of the log-convex minorant of a sequence $\\{M_α\\}_{α\\in\\mathbb{N}_0^d}$",
  "authors": [
    "Chiara Boiti",
    "David Jornet",
    "Alessandro Oliaro",
    "Gerhard Schindl"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "We give a simple construction of the log-convex minorant of a sequence $\\{M_\\alpha\\}_{\\alpha\\in\\mathbb{N}_0^d}$ and consequently extend to the $d$-dimensional case the well-known formula that relates a log-convex sequence $\\{M_p\\}_{p\\in\\mathbb{N}_0}$ to its associated function $\\omega_M$, that is $M_p=\\sup_{t>0}t^p\\exp(-\\omega_M(t))$. We show that in the more dimensional anisotropic case the classical log-convex condition $M_\\alpha^2\\leq M_{\\alpha-e_j}M_{\\alpha+e_j}$ is not sufficient: convexity as a function of more variables is needed (not only coordinate-wise). We finally obtain some applications to the inclusion of spaces of rapidly decreasing ultradifferentiable functions in the matrix weighted setting.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-20T14:57:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "40B05",
      "46A13",
      "46A45"
    ],
    "keywords": [
      "log-convex minorant",
      "dimensional anisotropic case",
      "dimensional case",
      "well-known formula",
      "log-convex sequence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}