{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "A9B7Z5SA9K"
    }
  ],
  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $f, g \\in L^1(\\R^n)$.\nLet $f * g$ be their [[convolution|2Z29JN3VES]], $(f * g)(x) = \\int_{\\R^n} f(y)\\, g(x - y)\\, dy$.\nLet $\\hat h(\\xi) = \\int_{\\R^n} h(x) e^{-2\\pi i x \\cdot \\xi}\\, dx$ denote the [[Fourier transform|DPVM34ADZV]] of $h \\in L^1(\\R^n)$.\nThen $f * g$ lies in [[$L^1(\\R^n)$|SASP14MA1Z]], and\n$$\\widehat{f * g}(\\xi) = \\hat f(\\xi)\\, \\hat g(\\xi) \\quad \\text{for every } \\xi \\in \\R^n.$$\n\n## Remarks\n\nThe proof is one application of [[Fubini's theorem|0R0T1VWH5X]].\nWrite $e^{-2\\pi i x \\cdot \\xi} = e^{-2\\pi i y \\cdot \\xi} e^{-2\\pi i (x - y) \\cdot \\xi}$ inside the double integral of $f(y) g(x - y) e^{-2\\pi i x \\cdot \\xi}$.\nThe double integral converges absolutely because $\\int\\int \\abs{f(y)} \\abs{g(x - y)}\\, dy\\, dx = \\norm{f}_1 \\norm{g}_1$.\nIntegrating first in $x$ and then in $y$ gives $\\hat f(\\xi) \\hat g(\\xi)$.\n\nExample: on $\\R$, let $f = g = \\mathbf{1}_{[-1/2, 1/2]}$.\nThen $\\hat f(\\xi) = \\sin(\\pi \\xi)/(\\pi \\xi)$, and $f * g$ is the triangle function $\\max(1 - \\abs{x}, 0)$.\nSo the Fourier transform of the triangle function is $\\sin^2(\\pi \\xi)/(\\pi \\xi)^2$.\n\nThere is a dual form: for $f, g$ in the Schwartz space, $\\widehat{f g} = \\hat f * \\hat g$.\nThe same identity holds for Fourier series, where the convolution on the circle becomes the product of Fourier coefficients.\n\n## Explanations\n\n### L1 human\n\nConvolution is a kind of moving weighted average of one function against another.\nOn the frequency side it becomes plain multiplication.\nSo to smooth a signal with a filter, one can multiply its spectrum by the spectrum of the filter.\n\n### L2 human\n\nThe theorem is why the Fourier transform solves linear equations with constant coefficients: such equations are convolution equations, and the transform makes them algebraic.\nIt also shows that $L^1(\\R^n)$ with convolution is a commutative algebra, and that $f \\mapsto \\hat f$ is an algebra homomorphism into bounded continuous functions with pointwise product.\nFast convolution of long sequences uses the discrete version of the theorem together with the fast Fourier transform.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "DPVM34ADZV",
      "2Z29JN3VES"
    ],
    "kind": "statement",
    "msc": [
      "42B10",
      "42A85",
      "42A38"
    ],
    "name": "Convolution theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q2638931",
      "wikipedia": "Convolution theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Convolution_theorem"
      }
    ],
    "tag": "QY8ETVDA1Z",
    "uses": [
      "DPVM34ADZV",
      "2Z29JN3VES",
      "SASP14MA1Z",
      "0R0T1VWH5X"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/QY8ETVDA1Z/",
  "links": [
    {
      "field": "about",
      "target": "DPVM34ADZV"
    },
    {
      "field": "about",
      "target": "2Z29JN3VES"
    },
    {
      "field": "uses",
      "target": "DPVM34ADZV"
    },
    {
      "field": "uses",
      "target": "2Z29JN3VES"
    },
    {
      "field": "uses",
      "target": "SASP14MA1Z"
    },
    {
      "field": "uses",
      "target": "0R0T1VWH5X"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/QY8ETVDA1Z.md",
  "msc": [
    "42B10",
    "42A85",
    "42A38"
  ],
  "name": "Convolution theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/QY8/QY8ETVDA1Z.md",
  "status": null,
  "tag": "QY8ETVDA1Z",
  "uri": "m2://QY8ETVDA1Z",
  "url": "https://tamnd.github.io/math2/wiki/convolution-theorem/",
  "writing": {
    "tag": "QY8ETVDA1Z",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}