{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $(G, K)$ be a [[Gelfand pair|W38ZDXDVYC]], and let $dk$ be the [[Haar measure|VY0HXD3GC4]] on $K$ of total mass $1$.\nA *spherical function* for $(G, K)$ is a [[continuous function|0P27W9RW0G]] $\\varphi \\colon G \\to \\C$ such that:\n\n1. $\\varphi(k_1 g k_2) = \\varphi(g)$ for all $g \\in G$ and $k_1, k_2 \\in K$;\n2. $\\varphi(e) = 1$;\n3. $\\int_K \\varphi(x k y) \\, dk = \\varphi(x) \\varphi(y)$ for all $x, y \\in G$.\n\n## Remarks\n\nExample: $G = \\R$ and $K = \\set{0}$, a Gelfand pair since $\\R$ is abelian.\nClause 1 is empty, and clause 3 reads $\\varphi(x + y) = \\varphi(x)\\varphi(y)$.\nSo the spherical functions are $\\varphi(x) = e^{s x}$ with $s \\in \\C$, and clause 2 holds since $e^0 = 1$.\nThe positive-definite ones are the unitary characters $e^{i \\xi x}$ with $\\xi \\in \\R$.\n\nNon-example: $\\varphi(x) = 2 e^{x}$ on $\\R$ fails clause 2, and also clause 3, since $\\varphi(x + y) \\ne \\varphi(x)\\varphi(y)$.\n\nFor $(\\mathrm{SO}(3), \\mathrm{SO}(2))$, the spherical functions are $g \\mapsto P_\\ell(\\cos \\theta(g))$, where $\\theta(g)$ is the angle between $g e_3$ and $e_3$.\nHere $P_\\ell$ is the Legendre polynomial of degree $\\ell \\ge 0$.\n\nEquivalent form: a continuous $K$-bi-invariant $\\varphi \\ne 0$ is spherical exactly when $f \\mapsto \\int_G f(g) \\varphi(g^{-1}) \\, dg$ is multiplicative on $C_c(K \\backslash G / K)$.\nIf $\\pi$ is an irreducible unitary representation with a unit $K$-fixed vector $v$, then $\\varphi(g) = \\ip{\\pi(g) v}{v}$ is a positive-definite spherical function.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "BKZNC76RC8",
    "kind": "definition",
    "msc": [
      "43A90",
      "22E30"
    ],
    "name": "Spherical function",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Zonal_spherical_function"
      }
    ],
    "tag": "KFW9841GQK",
    "uses": [
      "W38ZDXDVYC",
      "VY0HXD3GC4",
      "0P27W9RW0G"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/KFW9841GQK/",
  "links": [
    {
      "field": "defines",
      "target": "BKZNC76RC8"
    },
    {
      "field": "uses",
      "target": "W38ZDXDVYC"
    },
    {
      "field": "uses",
      "target": "VY0HXD3GC4"
    },
    {
      "field": "uses",
      "target": "0P27W9RW0G"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/KFW9841GQK.md",
  "msc": [
    "43A90",
    "22E30"
  ],
  "name": "Spherical function",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/KFW/KFW9841GQK.md",
  "status": null,
  "tag": "KFW9841GQK",
  "uri": "m2://KFW9841GQK",
  "url": "https://tamnd.github.io/math2/wiki/spherical-function-definition/",
  "writing": {
    "tag": "KFW9841GQK",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}