{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "640A1MTM4Z"
    }
  ],
  "body": "\n## Summary\n\nA paraproduct is the part of a product $fg$ in which the high frequencies of $f$ are multiplied by the low frequencies of $g$.\nUsing the [[Littlewood–Paley decomposition|D1PAT0K446]], every product splits as $fg = \\Pi_g f + \\Pi_f g + R(f, g)$, two paraproducts and a remainder in which both factors have comparable frequencies.\nParaproducts are bilinear operators that behave much better than the full product in many function spaces.\nThey appear in Bony's paradifferential calculus for nonlinear PDE and in the T(1) theorem, where they are the model non-convolution Calderón–Zygmund operators.\n\n## Explanations\n\n### L1 human\n\nWhen two functions are multiplied, their frequencies interact.\nIf $f$ oscillates fast and $g$ varies slowly, the product $fg$ oscillates at about the same frequencies as $f$, with an amplitude modulated by $g$.\nThe paraproduct $\\Pi_g f$ collects exactly these interactions, frequency band by frequency band.\nFor example, if $g$ is a constant $c$, then the paraproduct $\\Pi_g f$ is just $c f$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "42B25",
      "42B20",
      "35S50"
    ],
    "name": "Paraproduct",
    "same_as": {
      "wikidata": "Q17096244",
      "wikipedia": "Paraproduct"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Paraproduct"
      }
    ],
    "tag": "9RCFS9W90H"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/9RCFS9W90H/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/9RCFS9W90H.md",
  "msc": [
    "42B25",
    "42B20",
    "35S50"
  ],
  "name": "Paraproduct",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/9RC/9RCFS9W90H.md",
  "status": null,
  "tag": "9RCFS9W90H",
  "uri": "m2://9RCFS9W90H",
  "url": "https://tamnd.github.io/math2/wiki/paraproduct/",
  "writing": {
    "tag": "9RCFS9W90H",
    "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
      }
    ]
  }
}