{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\C^n$ be a domain, and let $L^2(\\Omega)$ be the [[Lebesgue space|SASP14MA1Z]] of square-integrable functions for Lebesgue measure $dV$.\nLet $A^2(\\Omega) \\subseteq L^2(\\Omega)$ be the [[Bergman space|2D619WTPYC]] of holomorphic functions in $L^2(\\Omega)$, which is a closed subspace.\nThe *Bergman projection* of $\\Omega$ is the [[orthogonal projection|0B8KZRT28M]]\n$$P \\colon L^2(\\Omega) \\to A^2(\\Omega).$$\nIt is given by\n$$P f(z) = \\int_\\Omega K(z, w) \\, f(w) \\, dV(w),$$\nwhere $K$ is the [[Bergman kernel|8RWDHW0CXC]] of $\\Omega$.\n\n## Remarks\n\nThe integral formula follows from the reproducing property: for $f \\in L^2(\\Omega)$, $Pf(z) = \\ip{Pf}{K(\\cdot, z)} = \\ip{f}{K(\\cdot, z)}$, since $K(\\cdot, z) \\in A^2(\\Omega)$ and $f - Pf$ is orthogonal to $A^2(\\Omega)$.\nExample: on the unit disc, the functions $z^k$, $k \\ge 0$, form an orthogonal basis of $A^2$, and $\\ip{\\bar z}{z^k} = \\int_{\\mathbb{D}} \\bar z \\, \\bar z^k \\, dV = 0$ for every $k$, so $P \\bar z = 0$.\nSimilarly $P \\abs{z}^2 = \\frac{1}{2}$, since $\\ip{\\abs{z}^2}{1} = \\int_{\\mathbb{D}} \\abs{z}^2 dV = \\frac{\\pi}{2}$ and $\\norm{1}^2 = \\pi$, and $\\abs{z}^2$ is orthogonal to $z^k$ for $k \\ge 1$.\nNon-example: the orthogonal projection of $L^2(\\Omega)$ onto the harmonic functions in $L^2(\\Omega)$ is a different operator, the harmonic Bergman projection; on the unit disc it fixes $\\bar z$.\n$P$ is self-adjoint, $P^2 = P$, and $\\norm{P} = 1$ when $A^2(\\Omega) \\ne \\set{0}$.\nOn smoothly bounded strongly pseudoconvex domains, $P$ maps $C^\\infty(\\bar\\Omega)$ into itself.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "6J1X63Z7QZ",
    "kind": "definition",
    "msc": [
      "32A36",
      "32A25",
      "30H20"
    ],
    "name": "Bergman projection",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Bergman_kernel"
      }
    ],
    "tag": "PHJ8DD2W0E",
    "uses": [
      "2D619WTPYC",
      "0B8KZRT28M",
      "8RWDHW0CXC",
      "SASP14MA1Z"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/PHJ8DD2W0E/",
  "links": [
    {
      "field": "defines",
      "target": "6J1X63Z7QZ"
    },
    {
      "field": "uses",
      "target": "2D619WTPYC"
    },
    {
      "field": "uses",
      "target": "0B8KZRT28M"
    },
    {
      "field": "uses",
      "target": "8RWDHW0CXC"
    },
    {
      "field": "uses",
      "target": "SASP14MA1Z"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/PHJ8DD2W0E.md",
  "msc": [
    "32A36",
    "32A25",
    "30H20"
  ],
  "name": "Bergman projection",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/PHJ/PHJ8DD2W0E.md",
  "status": null,
  "tag": "PHJ8DD2W0E",
  "uri": "m2://PHJ8DD2W0E",
  "url": "https://tamnd.github.io/math2/wiki/bergman-projection-definition/",
  "writing": {
    "tag": "PHJ8DD2W0E",
    "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
      }
    ]
  }
}