{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "140TG7W6A5"
    },
    {
      "field": "uses",
      "target": "199B14MWGC"
    },
    {
      "field": "uses",
      "target": "1JXFTBBK6D"
    },
    {
      "field": "uses",
      "target": "2M672PP7QR"
    },
    {
      "field": "uses",
      "target": "2R4H0RFQZZ"
    },
    {
      "field": "uses",
      "target": "375ZGE0V2W"
    },
    {
      "field": "uses",
      "target": "3CSEGRAD75"
    },
    {
      "field": "uses",
      "target": "3XC1C1NC0M"
    },
    {
      "field": "uses",
      "target": "41D1KJWM78"
    },
    {
      "field": "uses",
      "target": "4PDKJ8N1S9"
    },
    {
      "field": "uses",
      "target": "4ZY2D6C8A6"
    },
    {
      "field": "uses",
      "target": "5GX76EM5HS"
    },
    {
      "field": "uses",
      "target": "640A1MTM4Z"
    },
    {
      "field": "uses",
      "target": "6C1S8HD38Z"
    },
    {
      "field": "uses",
      "target": "8F1HPGFW4F"
    },
    {
      "field": "uses",
      "target": "9CE11G48JY"
    },
    {
      "field": "uses",
      "target": "A0MXE8YWZA"
    },
    {
      "field": "uses",
      "target": "A1X6T9ZD9E"
    },
    {
      "field": "uses",
      "target": "A4TRMA5V4V"
    },
    {
      "field": "uses",
      "target": "A7Q6581B1F"
    },
    {
      "field": "uses",
      "target": "A9B7Z5SA9K"
    },
    {
      "field": "uses",
      "target": "AT6JHPCXJ1"
    },
    {
      "field": "uses",
      "target": "B4TYSQQC38"
    },
    {
      "field": "uses",
      "target": "BMXDMHRXBG"
    },
    {
      "field": "uses",
      "target": "BRXKAGYEFQ"
    },
    {
      "field": "uses",
      "target": "BTPQHQNJVY"
    },
    {
      "field": "uses",
      "target": "BYKP1V2AQN"
    },
    {
      "field": "uses",
      "target": "CBRTFS0RVC"
    },
    {
      "field": "uses",
      "target": "CMMDQ13YEE"
    },
    {
      "field": "uses",
      "target": "CMXS9P2F13"
    },
    {
      "field": "uses",
      "target": "D7PH59MNTV"
    },
    {
      "field": "uses",
      "target": "EQBC3P901V"
    },
    {
      "field": "uses",
      "target": "EV25PMXVQT"
    },
    {
      "field": "uses",
      "target": "F5V7CXW30E"
    },
    {
      "field": "uses",
      "target": "FAMZ5Z5JK4"
    },
    {
      "field": "uses",
      "target": "FFCH15ZNZG"
    },
    {
      "field": "uses",
      "target": "FFM7Y61KRT"
    },
    {
      "field": "uses",
      "target": "GARGJYSGFG"
    },
    {
      "field": "uses",
      "target": "GETN4MCAAN"
    },
    {
      "field": "uses",
      "target": "H4DHJ7TS76"
    },
    {
      "field": "uses",
      "target": "HHNR31W3WD"
    },
    {
      "field": "uses",
      "target": "HN93ST5CBT"
    },
    {
      "field": "uses",
      "target": "HPZHRZ0502"
    },
    {
      "field": "uses",
      "target": "HXNJH6W23S"
    },
    {
      "field": "uses",
      "target": "J4TDK6Z4XA"
    },
    {
      "field": "uses",
      "target": "J7VA040MC3"
    },
    {
      "field": "uses",
      "target": "JJ5XQEVBZ4"
    },
    {
      "field": "uses",
      "target": "JKCFHKT13F"
    },
    {
      "field": "uses",
      "target": "M13GJHB8H9"
    },
    {
      "field": "uses",
      "target": "M435YDQWQ4"
    },
    {
      "field": "uses",
      "target": "MRBGM52H94"
    },
    {
      "field": "uses",
      "target": "MSMD2BS9C5"
    },
    {
      "field": "uses",
      "target": "NSXT1MYFY8"
    },
    {
      "field": "uses",
      "target": "P270MSQ0SZ"
    },
    {
      "field": "uses",
      "target": "PF3G0J5JD6"
    },
    {
      "field": "uses",
      "target": "PJN54XYB5T"
    },
    {
      "field": "uses",
      "target": "PJNV4KR00F"
    },
    {
      "field": "uses",
      "target": "PNZT35FFN2"
    },
    {
      "field": "uses",
      "target": "Q4HR7AF2R9"
    },
    {
      "field": "uses",
      "target": "Q5RN2FNX6Z"
    },
    {
      "field": "uses",
      "target": "QGGMQNFHJM"
    },
    {
      "field": "uses",
      "target": "QY8ETVDA1Z"
    },
    {
      "field": "uses",
      "target": "S1FB2K14NV"
    },
    {
      "field": "uses",
      "target": "SD89TZ1PHR"
    },
    {
      "field": "uses",
      "target": "SYP1ATSTTH"
    },
    {
      "field": "uses",
      "target": "T4Z8YQZKZV"
    },
    {
      "field": "uses",
      "target": "TAJTC44XC9"
    },
    {
      "field": "uses",
      "target": "V20376HXHD"
    },
    {
      "field": "uses",
      "target": "W3TAPJEX29"
    },
    {
      "field": "uses",
      "target": "WT7K28D1CD"
    },
    {
      "field": "uses",
      "target": "X17RYQ4DS2"
    },
    {
      "field": "uses",
      "target": "XJMGD9V64J"
    },
    {
      "field": "uses",
      "target": "XQN4N522WG"
    },
    {
      "field": "uses",
      "target": "XVVHKCYF2Z"
    },
    {
      "field": "uses",
      "target": "XWV6VS7BQM"
    },
    {
      "field": "uses",
      "target": "Z4ZFP6FNK2"
    },
    {
      "field": "uses",
      "target": "Z72JWTXM3W"
    },
    {
      "field": "uses",
      "target": "ZBXBQYKP5W"
    },
    {
      "field": "uses",
      "target": "ZE4QPBC1AF"
    },
    {
      "field": "uses",
      "target": "ZNEHMR2CCP"
    },
    {
      "field": "uses",
      "target": "ZS7HP7SGEP"
    },
    {
      "field": "uses",
      "target": "ZTZ1FRRFM8"
    },
    {
      "field": "uses",
      "target": "ZZVWHNFY6F"
    },
    {
      "field": "about",
      "target": "1JXFTBBK6D"
    },
    {
      "field": "about",
      "target": "3CSEGRAD75"
    },
    {
      "field": "about",
      "target": "3XC1C1NC0M"
    },
    {
      "field": "about",
      "target": "4ZY2D6C8A6"
    },
    {
      "field": "about",
      "target": "9CE11G48JY"
    },
    {
      "field": "about",
      "target": "A4TRMA5V4V"
    },
    {
      "field": "about",
      "target": "CBRTFS0RVC"
    },
    {
      "field": "about",
      "target": "EQBC3P901V"
    },
    {
      "field": "about",
      "target": "FFCH15ZNZG"
    },
    {
      "field": "about",
      "target": "FFM7Y61KRT"
    },
    {
      "field": "about",
      "target": "GARGJYSGFG"
    },
    {
      "field": "about",
      "target": "HHNR31W3WD"
    },
    {
      "field": "about",
      "target": "HPZHRZ0502"
    },
    {
      "field": "about",
      "target": "HXNJH6W23S"
    },
    {
      "field": "about",
      "target": "J7VA040MC3"
    },
    {
      "field": "about",
      "target": "QY8ETVDA1Z"
    },
    {
      "field": "about",
      "target": "S1FB2K14NV"
    },
    {
      "field": "about",
      "target": "SYP1ATSTTH"
    },
    {
      "field": "about",
      "target": "T4Z8YQZKZV"
    },
    {
      "field": "about",
      "target": "V20376HXHD"
    },
    {
      "field": "about",
      "target": "ZNEHMR2CCP"
    },
    {
      "field": "about",
      "target": "ZTZ1FRRFM8"
    },
    {
      "field": "about",
      "target": "ZZVWHNFY6F"
    },
    {
      "field": "defines",
      "target": "6V59T5KJBX"
    },
    {
      "field": "illustrates",
      "target": "A9B7Z5SA9K"
    }
  ],
  "body": "\n## Summary\n\nThe Fourier transform writes a function on $\\R^n$ as a superposition of plane waves $e^{2\\pi i x \\cdot \\xi}$ and records the amplitude of each frequency $\\xi$.\nIt turns derivatives into multiplication by polynomials in $\\xi$, and convolution into ordinary multiplication.\nSo it changes linear partial differential equations with constant coefficients into algebraic or ordinary differential equations.\nIt extends to $L^2$ functions and to [[tempered distributions|AES2T5JPFP]].\nSee the [[definition|6V59T5KJBX]].\n\n## Explanations\n\n### L1 human\n\nA sound is a mix of pure tones of different pitches.\nThe Fourier transform takes the sound as a function of time and tells how much of each pitch it contains.\nOn the frequency side, taking a derivative just multiplies each pitch by a factor proportional to its frequency.\nThis is why the transform solves equations such as the heat equation so easily.\nThe bell curve $e^{-\\pi x^2}$ is special: its transform is the same bell curve.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "42B10",
      "42A38",
      "35A22"
    ],
    "name": "Fourier transform",
    "same_as": {
      "wikidata": "Q6520159",
      "wikipedia": "Fourier transform"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Fourier_transform"
      }
    ],
    "tag": "DPVM34ADZV"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/DPVM34ADZV/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/DPVM34ADZV.md",
  "msc": [
    "42B10",
    "42A38",
    "35A22"
  ],
  "name": "Fourier transform",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/DPV/DPVM34ADZV.md",
  "status": null,
  "tag": "DPVM34ADZV",
  "uri": "m2://DPVM34ADZV",
  "url": "https://tamnd.github.io/math2/wiki/fourier-transform/",
  "writing": {
    "tag": "DPVM34ADZV",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}