{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a [[normed vector space|03J5RAKB30]], let $A \\subseteq X$ be nonempty, and let $x \\in X$.\nThe *error of best approximation* of $x$ from $A$ is the [[infimum|04KFTA34C5]]\n$$E(x, A) = \\inf_{a \\in A} \\norm{x - a}.$$\nIn particular, for a continuous $f \\colon [a, b] \\to \\R$ with the supremum norm and the set $\\mathcal{P}_n$ of real [[polynomial functions|1G0B73WPX8]] of degree at most $n$,\n$$E_n(f) = \\inf_{p \\in \\mathcal{P}_n} \\max_{a \\le x \\le b} \\abs{f(x) - p(x)}.$$\n\n## Remarks\n\nExample: for $f(x) = x^2$ on $[0, 1]$ one has $E_1(f) = 1/8$.\nThe line $p(x) = x - 1/8$ gives $\\abs{x^2 - x + 1/8} \\le 1/8$ on $[0, 1]$, and no line does better, because the error of $p$ takes the values $1/8, -1/8, 1/8$ at $0, 1/2, 1$.\n\nNon-example: the error of a particular approximation is in general larger.\nThe line $x$ has error $\\max_{[0, 1]} \\abs{x^2 - x} = 1/4$, which is not $E_1(f)$.\n\n$E(x, A) = 0$ if and only if $x$ lies in the closure of $A$.\nThe infimum equals $\\norm{x - a^*}$ when $a^*$ is a [[best approximation|7N4988HN3P]].\nSince $\\mathcal{P}_0 \\subseteq \\mathcal{P}_1 \\subseteq \\cdots$, the sequence $E_n(f)$ is nonincreasing.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "YDN0T4MEX0",
    "kind": "definition",
    "msc": [
      "41A50",
      "41A25",
      "41A10"
    ],
    "name": "Error of best approximation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Approximation_theory"
      }
    ],
    "tag": "C0AKF92CWH",
    "uses": [
      "03J5RAKB30",
      "04KFTA34C5",
      "7N4988HN3P",
      "1G0B73WPX8"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/C0AKF92CWH/",
  "links": [
    {
      "field": "defines",
      "target": "YDN0T4MEX0"
    },
    {
      "field": "uses",
      "target": "03J5RAKB30"
    },
    {
      "field": "uses",
      "target": "04KFTA34C5"
    },
    {
      "field": "uses",
      "target": "7N4988HN3P"
    },
    {
      "field": "uses",
      "target": "1G0B73WPX8"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/C0AKF92CWH.md",
  "msc": [
    "41A50",
    "41A25",
    "41A10"
  ],
  "name": "Error of best approximation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/C0A/C0AKF92CWH.md",
  "status": null,
  "tag": "C0AKF92CWH",
  "uri": "m2://C0AKF92CWH",
  "url": "https://tamnd.github.io/math2/wiki/error-of-best-approximation-definition/",
  "writing": {
    "tag": "C0AKF92CWH",
    "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
      }
    ]
  }
}