{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a [[Banach space|06GZ956657]] and let $L_n \\colon X \\to X$, $n \\ge 1$, be [[continuous linear maps|03TM135G0N]].\nLet $\\varphi(n) \u003e 0$ with $\\varphi(n) \\to 0$.\nThe sequence $(L_n)$ is *saturated* with *saturation order* $\\varphi(n)$ if both conditions hold.\n\n1. If $f \\in X$ and $\\norm{L_n f - f} = o(\\varphi(n))$, then $L_n f = f$ for all $n$.\n2. Some $f \\in X$ with $L_n f \\ne f$ for some $n$ satisfies $\\norm{L_n f - f} = O(\\varphi(n))$.\n\nThe *saturation class* is the set of $f \\in X$ with $\\norm{L_n f - f} = O(\\varphi(n))$, in [[big O notation|0TGV80E6A4]].\n\n## Remarks\n\nExample: the Bernstein operators $B_n$ on $C[0, 1]$ with the uniform norm are saturated with order $1/n$.\nFor $f(x) = x^2$ one has $B_n f(x) = x^2 + \\frac{x(1 - x)}{n}$, so $\\norm{B_n f - f} = \\frac{1}{4n}$, which is $O(1/n)$ but not $o(1/n)$.\nThe only $f$ with $\\norm{B_n f - f} = o(1/n)$ are the linear functions, which $B_n$ reproduces.\n\nNon-example: the Fourier partial sum operators $S_n$ on $L^2$ of the circle are not saturated for any order.\nA trigonometric polynomial $f$ of degree $k \\ge 1$ has $S_n f = f$ for $n \\ge k$, so $\\norm{S_n f - f} = o(\\varphi(n))$, but $S_0 f \\ne f$, so clause 1 fails.\n\nThe Fejér means of Fourier series are saturated with order $1/n$ in $C$ of the circle, and only constants are approximated with error $o(1/n)$.\nSaturation theorems typically identify the saturation class with a smoothness class, such as functions with Lipschitz first derivative for Bernstein operators.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "3QWJB8GHDE",
    "kind": "definition",
    "msc": [
      "41A40",
      "41A35",
      "41A36"
    ],
    "name": "Saturation of approximation methods",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Approximation_theory"
      }
    ],
    "tag": "54BD10ZZDN",
    "uses": [
      "06GZ956657",
      "03TM135G0N",
      "0TGV80E6A4"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/54BD10ZZDN/",
  "links": [
    {
      "field": "defines",
      "target": "3QWJB8GHDE"
    },
    {
      "field": "uses",
      "target": "06GZ956657"
    },
    {
      "field": "uses",
      "target": "03TM135G0N"
    },
    {
      "field": "uses",
      "target": "0TGV80E6A4"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/54BD10ZZDN.md",
  "msc": [
    "41A40",
    "41A35",
    "41A36"
  ],
  "name": "Saturation of approximation methods",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/54B/54BD10ZZDN.md",
  "status": null,
  "tag": "54BD10ZZDN",
  "uri": "m2://54BD10ZZDN",
  "url": "https://tamnd.github.io/math2/wiki/saturation-of-approximation-methods-definition/",
  "writing": {
    "tag": "54BD10ZZDN",
    "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
      }
    ]
  }
}