{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "2AKFNAGF1N"
    },
    {
      "field": "uses",
      "target": "3EQXGGB849"
    },
    {
      "field": "uses",
      "target": "Q6KCF4688J"
    },
    {
      "field": "uses",
      "target": "XKAWV4CCCB"
    },
    {
      "field": "about",
      "target": "Q6KCF4688J"
    }
  ],
  "body": "\n## Summary\n\nA positive linear operator is a [[linear map|0NRYHJPQG2]] between ordered vector spaces that sends nonnegative elements to nonnegative elements.\nOn spaces of functions, it maps nonnegative functions to nonnegative functions.\nBernstein polynomials, convolution with nonnegative kernels and integration are all positive linear operators.\nTheir positivity gives strong convergence tests, such as Korovkin's theorem.\n\n## Explanations\n\n### L1 human\n\nA positive operator never turns something nonnegative into something negative.\nAveraging is the typical example: an average of nonnegative numbers is nonnegative.\nSo any operator that replaces $f(x)$ by a weighted average of values of $f$, with nonnegative weights, is positive.\nDifferentiation is not positive: $1 - x$ is nonnegative on $[0, 1]$, but its derivative is $-1$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "47B65",
      "41A36",
      "46A40"
    ],
    "name": "Positive linear operator",
    "same_as": {
      "wikidata": "Q96399850",
      "wikipedia": "Positive linear operator"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Positive_linear_operator"
      }
    ],
    "tag": "AQDXWH4812"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/AQDXWH4812/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/AQDXWH4812.md",
  "msc": [
    "47B65",
    "41A36",
    "46A40"
  ],
  "name": "Positive linear operator",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/AQD/AQDXWH4812.md",
  "status": null,
  "tag": "AQDXWH4812",
  "uri": "m2://AQDXWH4812",
  "url": "https://tamnd.github.io/math2/wiki/positive-linear-operator/",
  "writing": {
    "tag": "AQDXWH4812",
    "kind": "concept",
    "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
      }
    ]
  }
}