{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "6Y7NJCXXTN"
    }
  ],
  "body": "\n## Summary\n\nA convolution of an [[integral transform|YCK709QB06]] $T$ is a bilinear product $\\circledast$ of functions that $T$ turns into the pointwise product: $T(f \\circledast g) = Tf \\cdot Tg$.\nThe usual [[convolution on the line|ERP5737VZ1]] plays this role for the Fourier transform.\nThe [[convolution on the half-line|FCKX8KFVGN]] plays it for the Laplace transform, and the [[Mellin convolution|A2PMJG7J68]] for the Mellin transform.\nFinding such a product for a given transform gives it a convolution theorem, which is what makes it useful for solving integral and differential equations.\n\n## Explanations\n\n### L1 human\n\nMany transforms are useful because they turn a hard operation into multiplication.\nFor each transform one can ask which operation on the original functions becomes multiplication on the transform side.\nThat operation is the convolution of the transform.\nFor the Fourier transform it is the familiar convolution; for other transforms it is a different integral, adapted to the kernel of the transform.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "44A35",
      "44A05"
    ],
    "name": "Generalized convolution of an integral transform",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Convolution"
      }
    ],
    "tag": "EAP9N3VEBV"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/EAP9N3VEBV/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/EAP9N3VEBV.md",
  "msc": [
    "44A35",
    "44A05"
  ],
  "name": "Generalized convolution of an integral transform",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/EAP/EAP9N3VEBV.md",
  "status": null,
  "tag": "EAP9N3VEBV",
  "uri": "m2://EAP9N3VEBV",
  "url": "https://tamnd.github.io/math2/wiki/generalized-convolution-of-an-integral-transform/",
  "writing": {
    "tag": "EAP9N3VEBV",
    "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
      }
    ]
  }
}