{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $a \u003c b$ be real numbers and let $m, n \\ge 0$ be integers.\nLet $R_{m,n}$ be the set of [[rational functions|0WCZH4AFMT]] $r = p/q$ with real polynomials $p$ and $q$ such that $\\deg p \\le m$ and $\\deg q \\le n$.\nIn addition, $q$ has no zero on $[a, b]$.\nLet $f \\colon [a, b] \\to \\R$ be a [[continuous function|0X3WGX95FE]].\nA *rational approximation* of type $(m, n)$ to $f$ is an element $r$ of $R_{m,n}$ used in place of $f$.\nIts error is $\\norm{f - r}_\\infty = \\max_{x \\in [a, b]} \\abs{f(x) - r(x)}$.\nThe *error of best rational approximation* is\n$$E_{m,n}(f) = \\inf_{r \\in R_{m,n}} \\norm{f - r}_\\infty.$$\n\n## Remarks\n\nFor $n = 0$ this is polynomial approximation, and $E_{m,0}(f)$ is the error of [[best approximation|7N4988HN3P]] by polynomials of degree at most $m$.\nFor continuous $f$ on $[a, b]$, a best rational approximation in $R_{m,n}$ exists and is unique, and Chebyshev's equioscillation theorem characterizes it.\n$R_{m,n}$ is not a vector space for $n \\ge 1$, since a sum of two of its elements may need numerator degree $m + n$ and denominator degree $2n$.\nThis is why the theory differs from the linear case.\nExample: $r(x) = 1/(1 + x^2)$ lies in $R_{0,2}$ on any interval, and $E_{0,2}(r) = 0$.\nNon-example: on $[-1, 1]$, $1/x$ is not an element of $R_{0,1}$, since its denominator vanishes at $0 \\in [-1, 1]$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "EHFQK1WXKS",
    "kind": "definition",
    "msc": [
      "41A20",
      "41A21",
      "41A50"
    ],
    "name": "Rational approximation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Rational_function"
      }
    ],
    "tag": "76JD0ZG3B1",
    "uses": [
      "0WCZH4AFMT",
      "7N4988HN3P",
      "0X3WGX95FE"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/76JD0ZG3B1/",
  "links": [
    {
      "field": "defines",
      "target": "EHFQK1WXKS"
    },
    {
      "field": "uses",
      "target": "0WCZH4AFMT"
    },
    {
      "field": "uses",
      "target": "7N4988HN3P"
    },
    {
      "field": "uses",
      "target": "0X3WGX95FE"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/76JD0ZG3B1.md",
  "msc": [
    "41A20",
    "41A21",
    "41A50"
  ],
  "name": "Rational approximation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/76J/76JD0ZG3B1.md",
  "status": null,
  "tag": "76JD0ZG3B1",
  "uri": "m2://76JD0ZG3B1",
  "url": "https://tamnd.github.io/math2/wiki/rational-approximation-definition/",
  "writing": {
    "tag": "76JD0ZG3B1",
    "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
      }
    ]
  }
}