{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\mathbb{D}$ be the open [[unit disc|8G5ABXRK85]].\nLet $\\varphi$ be a nonnegative measurable function on the unit circle such that $\\log \\varphi$ is integrable with respect to [[Lebesgue measure|2Z2QVWZ262]].\nLet $c \\in \\C$ with $\\abs{c} = 1$.\nThe *outer function* with modulus $\\varphi$ and constant $c$ is the function $F \\colon \\mathbb{D} \\to \\C$ given by\n$$F(z) = c \\exp\\left(\\frac{1}{2\\pi} \\int_0^{2\\pi} \\frac{e^{it} + z}{e^{it} - z} \\log \\varphi(e^{it}) \\, dt\\right).$$\nA holomorphic function on $\\mathbb{D}$ is *outer* if it is of this form for some $\\varphi$ and $c$.\n\n## Remarks\n\n$F$ is [[holomorphic|0DG0Z72CQD]] and has no zeros, and $\\log \\abs{F}$ is the [[Poisson integral|TCYSWCTYYM]] of $\\log \\varphi$.\nIts radial limits satisfy $\\abs{F^*} = \\varphi$ almost everywhere.\nFor $0 \u003c p \\le \\infty$, $F$ lies in the Hardy space $H^p$ if and only if $\\varphi \\in L^p$ ([[Lp space|SASP14MA1Z]]).\n\nExample: $F(z) = 2 + z$ is outer.\nIt is holomorphic and has no zeros on a neighbourhood of the closed disc, so $\\log \\abs{F}$ is harmonic and continuous up to the circle.\nHence $\\log \\abs{F}$ equals the Poisson integral of $\\log \\varphi$, where $\\varphi = \\abs{2 + e^{it}}$.\nSo $F$ and the outer function with modulus $\\varphi$ have the same modulus and no zeros; their quotient is a holomorphic function of constant modulus $1$, hence a constant.\n\nNon-example: $f(z) = z$ is not outer.\nIts boundary modulus is $\\varphi = 1$, and the outer functions with this modulus are the constants of modulus $1$, which differ from $z$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "KGJD4A4ATY",
    "kind": "definition",
    "msc": [
      "30H10",
      "30J99"
    ],
    "name": "Outer function",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hardy_space"
      }
    ],
    "tag": "T97B2GX3JJ",
    "uses": [
      "8G5ABXRK85",
      "0DG0Z72CQD",
      "TCYSWCTYYM"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/T97B2GX3JJ/",
  "links": [
    {
      "field": "defines",
      "target": "KGJD4A4ATY"
    },
    {
      "field": "uses",
      "target": "8G5ABXRK85"
    },
    {
      "field": "uses",
      "target": "0DG0Z72CQD"
    },
    {
      "field": "uses",
      "target": "TCYSWCTYYM"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/T97B2GX3JJ.md",
  "msc": [
    "30H10",
    "30J99"
  ],
  "name": "Outer function",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/T97/T97B2GX3JJ.md",
  "status": null,
  "tag": "T97B2GX3JJ",
  "uri": "m2://T97B2GX3JJ",
  "url": "https://math2.tamnd.com/wiki/outer-function-definition/",
  "writing": {
    "tag": "T97B2GX3JJ",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}