{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $n \\ge 2$ and $m \\ge 2$ be integers, and write $\\abs{x}^2 = x_1^2 + \\cdots + x_n^2$ for $x \\in \\R^n$.\nLet $p$ be a [[homogeneous polynomial|2C3PAQ7H50]] of degree $m$ in $x_1, \\dots, x_n$ with real or complex coefficients.\nThen there are unique homogeneous polynomials $h$ of degree $m$ and $q$ of degree $m - 2$ such that $h$ is [[harmonic|MVYFSYMECG]] and\n$$p = h + \\abs{x}^2 q.$$\nRepeating this with $q$ gives the unique expansion\n$$p = h_m + \\abs{x}^2 h_{m-2} + \\abs{x}^4 h_{m-4} + \\cdots.$$\nHere each $h_j$ is a homogeneous harmonic polynomial of degree $j$, and the sum ends at $j = 0$ or $j = 1$.\n\n## Remarks\n\nProof idea: on the space $P_m$ of homogeneous polynomials of degree $m$, use the inner product $\\langle p, q \\rangle = p(\\partial) \\bar q$, where $p(\\partial)$ replaces each $x_i$ by $\\partial / \\partial x_i$.\nFor this inner product, multiplication by $\\abs{x}^2$ from $P_{m-2}$ to $P_m$ is the adjoint of the [[Laplacian|0E0B6RA8K0]] from $P_m$ to $P_{m-2}$.\nSo the harmonic polynomials of degree $m$, which form the kernel of the Laplacian, are the orthogonal complement of $\\abs{x}^2 P_{m-2}$ in $P_m$.\nExample: $x_1^2 = \\left(x_1^2 - \\tfrac{1}{n}\\abs{x}^2\\right) + \\abs{x}^2 \\cdot \\tfrac{1}{n}$, and the first term is harmonic since its Laplacian is $2 - \\tfrac{2n}{n} = 0$.\nRestricted to the unit sphere, where $\\abs{x}^2 = 1$, the expansion shows that every polynomial is a finite sum of spherical harmonics.\n\n## Explanations\n\n### L1 human\n\nEvery homogeneous polynomial splits into a harmonic part plus a multiple of $\\abs{x}^2$.\nRepeating the split writes the polynomial as harmonic pieces times powers of $\\abs{x}^2$.\nOn the unit sphere the powers of $\\abs{x}^2$ equal $1$, so only harmonic pieces remain.\n\n### L2 human\n\nThe decomposition gives the dimension of the space of homogeneous harmonic polynomials of degree $m$: it is $\\dim P_m - \\dim P_{m-2}$, which is $2m + 1$ in $\\R^3$.\nIt shows that the restrictions of polynomials to the sphere are dense in the continuous functions on the sphere, and that they are sums of spherical harmonics.\nThis leads to the decomposition of $L^2$ of the sphere into spherical harmonics, which underlies the [[expansion in spherical harmonics|YTPE0GBKWW]].\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "MVYFSYMECG"
    ],
    "kind": "statement",
    "msc": [
      "31B05",
      "33C55"
    ],
    "name": "Fischer decomposition",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Harmonic_polynomial"
      }
    ],
    "tag": "JWNRB34ZGV",
    "uses": [
      "2C3PAQ7H50",
      "MVYFSYMECG",
      "0E0B6RA8K0"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/JWNRB34ZGV/",
  "links": [
    {
      "field": "about",
      "target": "MVYFSYMECG"
    },
    {
      "field": "uses",
      "target": "2C3PAQ7H50"
    },
    {
      "field": "uses",
      "target": "MVYFSYMECG"
    },
    {
      "field": "uses",
      "target": "0E0B6RA8K0"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/JWNRB34ZGV.md",
  "msc": [
    "31B05",
    "33C55"
  ],
  "name": "Fischer decomposition",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/JWN/JWNRB34ZGV.md",
  "status": null,
  "tag": "JWNRB34ZGV",
  "uri": "m2://JWNRB34ZGV",
  "url": "https://tamnd.github.io/math2/wiki/fischer-decomposition/",
  "writing": {
    "tag": "JWNRB34ZGV",
    "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
      }
    ]
  }
}