{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $\\mathbb{D} = \\set{z \\in \\C : \\abs{z} \u003c 1}$ and $\\mathbb{T} = \\set{\\zeta \\in \\C : \\abs{\\zeta} = 1}$.\n\n1. Let $u \\colon \\mathbb{D} \\to \\R$ be a [[harmonic function|0DEB0JMN0W]] with $u \\ge 0$. Then there is a unique finite positive Borel measure $\\mu$ on $\\mathbb{T}$ such that for every $z \\in \\mathbb{D}$\n$$u(z) = \\int_{\\mathbb{T}} \\frac{1 - \\abs{z}^2}{\\abs{\\zeta - z}^2}\\, d\\mu(\\zeta).$$\nIts total mass is $\\mu(\\mathbb{T}) = u(0)$.\n2. Let $f$ be a [[holomorphic function|0NAHXR1FRH]] on $\\mathbb{D}$ with $\\operatorname{Re} f \\ge 0$. Then, with $\\mu$ the measure of part 1 for $u = \\operatorname{Re} f$, for every $z \\in \\mathbb{D}$\n$$f(z) = \\int_{\\mathbb{T}} \\frac{\\zeta + z}{\\zeta - z}\\, d\\mu(\\zeta) + i \\operatorname{Im} f(0).$$\n\n## Remarks\n\nThe integrand in part 1 is the [[Poisson kernel|W19567AVYG]].\nFor $0 \u003c r \u003c 1$, the measures $\\mu_r = u(r\\zeta)\\, \\frac{\\abs{d\\zeta}}{2\\pi}$ on $\\mathbb{T}$ are positive with total mass $u(0)$, by the mean value property.\nThe Poisson formula on the disc of radius $r$ expresses $u(rz)$ through $\\mu_r$; a weak-star limit of the $\\mu_r$ as $r \\to 1$ gives $\\mu$.\nPart 2 follows from part 1 as in the [[Schwarz integral formula|61WT6CE08W]].\nExample: for the Dirac measure at $1$, the formula of part 1 gives $u(z) = (1 - \\abs{z}^2)/\\abs{1 - z}^2$, a positive harmonic function that is unbounded near $1$, and part 2 gives $f(z) = (1 + z)/(1 - z)$.\nThe hypothesis $u \\ge 0$ is needed.\nThe function $u(z) = \\operatorname{Re} \\frac{(1 + z)^2}{(1 - z)^2}$ is harmonic but takes both signs.\nIt is not the Poisson integral of any finite signed measure, because it grows too fast near $1$.\n\n## Explanations\n\n### L1 human\n\nA nonnegative harmonic function on the disc is a weighted average of the simplest ones, the Poisson kernels with their pole at a single boundary point.\nThe weights form a measure on the boundary circle.\nSo every such function is built from a mass distribution on the circle, and the mass distribution is unique.\n\n### L2 human\n\nThe theorem identifies the cone of nonnegative harmonic functions on the disc with the cone of finite positive measures on the circle.\nIn the holomorphic form it describes all functions with nonnegative real part, the Carathéodory class, which is used in coefficient problems for [[univalent functions|R91E83SDBD]].\nIt also underlies spectral representations in operator theory, such as the moment problem and the spectral theorem for unitary operators.\nTogether with [[Harnack's inequality|YNH832D50W]] it controls the size of positive harmonic functions.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "about": [
      "0DEB0JMN0W",
      "TCYSWCTYYM"
    ],
    "kind": "statement",
    "msc": [
      "31A05",
      "31A20",
      "30H99"
    ],
    "name": "Herglotz representation theorem",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Positive_harmonic_function"
      }
    ],
    "tag": "9ZBAX4ZPX0",
    "uses": [
      "0DEB0JMN0W",
      "W19567AVYG",
      "0NAHXR1FRH"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/9ZBAX4ZPX0/",
  "links": [
    {
      "field": "about",
      "target": "0DEB0JMN0W"
    },
    {
      "field": "about",
      "target": "TCYSWCTYYM"
    },
    {
      "field": "uses",
      "target": "0DEB0JMN0W"
    },
    {
      "field": "uses",
      "target": "W19567AVYG"
    },
    {
      "field": "uses",
      "target": "0NAHXR1FRH"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/9ZBAX4ZPX0.md",
  "msc": [
    "31A05",
    "31A20",
    "30H99"
  ],
  "name": "Herglotz representation theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/9ZB/9ZBAX4ZPX0.md",
  "status": null,
  "tag": "9ZBAX4ZPX0",
  "uri": "m2://9ZBAX4ZPX0",
  "url": "https://math2.tamnd.com/wiki/herglotz-representation-theorem/",
  "writing": {
    "tag": "9ZBAX4ZPX0",
    "kind": "statement",
    "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
      }
    ]
  }
}