{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $\\mathcal{S}(\\R^n)$ be the [[Schwartz space|CG9JEATW3Q]] with its Fréchet topology.\nLet $\\mathcal{F} f(\\xi) = \\int_{\\R^n} f(x) e^{-2\\pi i x \\cdot \\xi}\\, dx$ be the [[Fourier transform|DPVM34ADZV]].\nThen $\\mathcal{F}$ maps $\\mathcal{S}(\\R^n)$ onto $\\mathcal{S}(\\R^n)$, it is a linear bijection, and both $\\mathcal{F}$ and its inverse are continuous.\nThe inverse is the [[inverse Fourier transform|JB8QMT8DCH]]\n$$\\mathcal{F}^{-1} g(x) = \\int_{\\R^n} g(\\xi)\\, e^{2\\pi i x \\cdot \\xi}\\, d\\xi.$$\n\n## Remarks\n\nBy the [[rules for differentiation and multiplication|HHNR31W3WD]], $\\xi^\\alpha \\partial^\\beta \\hat f$ is, up to constants, the transform of $\\partial^\\alpha (x^\\beta f)$.\nThat function is in $\\mathcal{S}(\\R^n) \\subset L^1(\\R^n)$, so its transform is bounded, which shows $\\hat f \\in \\mathcal{S}(\\R^n)$.\nThe same estimates, with a factor $(1 + \\abs{x})^{-n-1}$ to control the integral, give continuity.\nBijectivity is the inversion formula $\\mathcal{F}^{-1} \\mathcal{F} f = f$, proved by pairing with a Gaussian of shrinking width.\nAlso $\\mathcal{F}^4 f = f$, since $\\mathcal{F}^2 f(x) = f(-x)$.\n\nExample: the Gaussian $e^{-\\pi \\abs{x}^2}$ is a Schwartz function and is fixed by $\\mathcal{F}$.\n\nOn $\\mathcal{S}(\\R^n)$ the transform also preserves the $L^2$ norm, $\\int \\abs{\\hat f}^2 = \\int \\abs{f}^2$.\nSince $\\mathcal{S}(\\R^n)$ is dense in $L^2(\\R^n)$, this is how the transform is extended to a unitary map of $L^2(\\R^n)$.\nBy duality it also extends to tempered distributions.\nThe space $L^1(\\R^n)$ has no such property: the transform of an integrable function need not be integrable.\n\n## Explanations\n\n### L1 human\n\nSchwartz functions are smooth and decay fast, and the Fourier transform turns smoothness into decay and decay into smoothness.\nSo it sends this space to itself, and nothing is lost: every Schwartz function is the transform of exactly one other.\n\n### L2 human\n\nThis is the cleanest setting for Fourier analysis on $\\R^n$: all formulas (inversion, Plancherel, the convolution theorem, differentiation rules) hold on $\\mathcal{S}(\\R^n)$ without any conditions.\nResults on $L^p$ spaces are then obtained by density, and results on tempered distributions by duality.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "DPVM34ADZV",
      "CG9JEATW3Q"
    ],
    "kind": "statement",
    "msc": [
      "42B10",
      "46F05"
    ],
    "name": "Fourier transform on Schwartz space",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Schwartz_space"
      }
    ],
    "tag": "1JXFTBBK6D",
    "uses": [
      "DPVM34ADZV",
      "CG9JEATW3Q",
      "HHNR31W3WD",
      "JB8QMT8DCH"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/1JXFTBBK6D/",
  "links": [
    {
      "field": "about",
      "target": "DPVM34ADZV"
    },
    {
      "field": "about",
      "target": "CG9JEATW3Q"
    },
    {
      "field": "uses",
      "target": "DPVM34ADZV"
    },
    {
      "field": "uses",
      "target": "CG9JEATW3Q"
    },
    {
      "field": "uses",
      "target": "HHNR31W3WD"
    },
    {
      "field": "uses",
      "target": "JB8QMT8DCH"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/1JXFTBBK6D.md",
  "msc": [
    "42B10",
    "46F05"
  ],
  "name": "Fourier transform on Schwartz space",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/1JX/1JXFTBBK6D.md",
  "status": null,
  "tag": "1JXFTBBK6D",
  "uri": "m2://1JXFTBBK6D",
  "url": "https://tamnd.github.io/math2/wiki/fourier-transform-on-schwartz-space/",
  "writing": {
    "tag": "1JXFTBBK6D",
    "kind": "statement",
    "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
      }
    ]
  }
}