{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "ZGH48BG36Z"
    }
  ],
  "body": "\n## Summary\n\nThe Faber polynomials of a compact set $K$ in the plane are polynomials $F_0, F_1, F_2, \\dots$, with $F_n$ of degree $n$.\nThey are built from the [[conformal map|0VHNKP1N3K]] of the complement of $K$ onto the exterior of the unit disc.\nThey play for $K$ the role that the powers $z^n$ play for a disc.\nFunctions holomorphic on a neighbourhood of $K$ can be expanded in Faber series $\\sum_n b_n F_n(z)$.\nThey are a standard tool of polynomial approximation and of geometric function theory.\n\n## Explanations\n\n### L1 human\n\nOn a disc centred at $0$, the natural building blocks for holomorphic functions are the powers $z^n$.\nFor another shape, such as an ellipse or a segment, one first straightens its outside into the outside of a circle by a conformal map $\\Phi$.\nThe $n$-th Faber polynomial is the polynomial part of $\\Phi(z)^n$ near infinity.\nFor the closed unit disc, $\\Phi(z) = z$ and $F_n(z) = z^n$.\nFor the segment $[-1, 1]$ one gets twice the Chebyshev polynomials.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "kind": "concept",
    "msc": [
      "30E10",
      "30C10",
      "41A10"
    ],
    "name": "Faber polynomials",
    "same_as": {
      "wikidata": "Q4298932",
      "wikipedia": "Faber polynomials"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Faber_polynomials"
      }
    ],
    "tag": "PVTA9RQKWM"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/PVTA9RQKWM/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/PVTA9RQKWM.md",
  "msc": [
    "30E10",
    "30C10",
    "41A10"
  ],
  "name": "Faber polynomials",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/PVT/PVTA9RQKWM.md",
  "status": null,
  "tag": "PVTA9RQKWM",
  "uri": "m2://PVTA9RQKWM",
  "url": "https://math2.tamnd.com/wiki/faber-polynomials/",
  "writing": {
    "tag": "PVTA9RQKWM",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}