{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "QQ6DA5QKXB"
    }
  ],
  "body": "\n## Summary\n\nThe Funk transform sends a function on the sphere $S^2$ to the function that gives its integral over each great circle.\nIt is the spherical analogue of the [[Radon transform|VJNZJ352PC]], with great circles in place of lines.\nIt kills odd functions, and it is injective on even continuous functions.\nOn [[spherical harmonics|3F32Y21HP4]] it acts by multiplication by explicit constants, which gives its inversion.\n\n## Explanations\n\n### L1 human\n\nThink of a function on the surface of the Earth, such as temperature.\nFor every great circle, such as the equator or a meridian circle through both poles, add up the temperature along it.\nThe Funk transform records these totals, indexed by the pole of each great circle.\nA function with $f(-x) = -f(x)$ gives total $0$ on every great circle, so it is invisible.\nFor functions with $f(-x) = f(x)$, the totals determine the function.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "44A12",
      "53C65",
      "33C55"
    ],
    "name": "Funk transform",
    "same_as": {
      "wikidata": "Q5509245",
      "wikipedia": "Funk transform"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Funk_transform"
      }
    ],
    "specializes": [
      "YCK709QB06"
    ],
    "tag": "WQPTRZYY0Y"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/WQPTRZYY0Y/",
  "links": [
    {
      "field": "specializes",
      "target": "YCK709QB06"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/WQPTRZYY0Y.md",
  "msc": [
    "44A12",
    "53C65",
    "33C55"
  ],
  "name": "Funk transform",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/WQP/WQPTRZYY0Y.md",
  "status": null,
  "tag": "WQPTRZYY0Y",
  "uri": "m2://WQPTRZYY0Y",
  "url": "https://tamnd.github.io/math2/wiki/funk-transform/",
  "writing": {
    "tag": "WQPTRZYY0Y",
    "kind": "concept",
    "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
      }
    ]
  }
}