{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $S$ be a set, let $f \\colon S \\to \\R$ be a [[function|0EPAT51DXQ]], and let $A$ be a set of functions $S \\to \\R$.\nFor $\\varepsilon \u003e 0$, a function $g \\in A$ *approximates $f$ uniformly within $\\varepsilon$* if\n$$\\sup_{x \\in S} \\abs{f(x) - g(x)} \\le \\varepsilon.$$\nThe function $f$ *can be uniformly approximated by $A$* if such a $g \\in A$ exists for every $\\varepsilon \u003e 0$.\nEquivalently, there is a sequence $g_n \\in A$ that [[converges uniformly|08FPABDQ4C]] to $f$ on $S$.\n\n## Remarks\n\nExample: on $S = [0, 1]$, the function $e^x$ can be uniformly approximated by polynomials.\nBy Taylor's theorem, $\\abs{e^x - \\sum_{k=0}^{n} x^k / k!} \\le e / (n + 1)!$ for $x \\in [0, 1]$, and the bound tends to $0$.\n\nNon-example: on $S = \\R$, the function $e^x$ cannot be uniformly approximated by polynomials.\nFor each polynomial $p$, the difference $e^x - p(x)$ tends to $\\infty$ as $x \\to \\infty$, so the supremum is infinite.\n\nFor bounded functions, the quantity $\\sup_S \\abs{f - g}$ is the distance in the supremum norm, and uniform approximation is [[function approximation|VNWH2KADN7]] in that norm.\nComplex-valued functions are treated in the same way.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "XAARVNKXF1",
    "kind": "definition",
    "msc": [
      "41A65",
      "41A50",
      "41A10"
    ],
    "name": "Uniform approximation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Uniform_norm"
      }
    ],
    "tag": "TFB4BYRZ1W",
    "uses": [
      "VNWH2KADN7",
      "08FPABDQ4C",
      "0EPAT51DXQ"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/TFB4BYRZ1W/",
  "links": [
    {
      "field": "defines",
      "target": "XAARVNKXF1"
    },
    {
      "field": "uses",
      "target": "VNWH2KADN7"
    },
    {
      "field": "uses",
      "target": "08FPABDQ4C"
    },
    {
      "field": "uses",
      "target": "0EPAT51DXQ"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/TFB4BYRZ1W.md",
  "msc": [
    "41A65",
    "41A50",
    "41A10"
  ],
  "name": "Uniform approximation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/TFB/TFB4BYRZ1W.md",
  "status": null,
  "tag": "TFB4BYRZ1W",
  "uri": "m2://TFB4BYRZ1W",
  "url": "https://tamnd.github.io/math2/wiki/uniform-approximation-definition/",
  "writing": {
    "tag": "TFB4BYRZ1W",
    "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
      }
    ]
  }
}