{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $1 \\le p \u003c \\infty$ and let $f \\in L^p(\\R)$, a function in the [[$L^p$ space|SASP14MA1Z]] of $\\R$.\nThe *Hilbert transform* of $f$ is the function\n$$Hf(x) = \\frac{1}{\\pi}\\, \\lim_{\\varepsilon \\to 0^+} \\int_{\\abs{x - y} \u003e \\varepsilon} \\frac{f(y)}{x - y}\\, dy,$$\nthe [[Cauchy principal value|0SYR10HD4R]] of $\\frac{1}{\\pi} \\int_\\R \\frac{f(y)}{x - y}\\, dy$.\nThe limit exists for almost every $x \\in \\R$.\n\n## Remarks\n\nExample: for $f(x) = \\frac{1}{1 + x^2}$, one has $Hf(x) = \\frac{x}{1 + x^2}$.\nCheck: $F(z) = \\frac{i}{z + i}$ is holomorphic on the upper half-plane and tends to $0$ at infinity.\nOn $\\R$ it equals $\\frac{1}{1 + x^2} + i \\frac{x}{1 + x^2}$, so its imaginary part is the [[harmonic conjugate|050F2FMTKF]] of its real part.\n\nNon-example: the integral is singular at $y = x$, so the principal value is needed.\nThe integral $\\int_\\R \\frac{f(y)}{x - y}\\, dy$ diverges absolutely at every $x$ near which $f$ is continuous and nonzero.\n\nOn the Fourier transform side, $\\widehat{Hf}(\\xi) = -i \\sgn(\\xi)\\, \\hat f(\\xi)$ for $f \\in L^2(\\R)$.\nHence $H$ is an isometry of $L^2(\\R)$ with $H^2 = -\\id$.\nFor $1 \u003c p \u003c \\infty$, $H$ is a bounded operator on $L^p(\\R)$, a theorem of Marcel Riesz; for $p = 1$ it is only of weak type $(1, 1)$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "NSCEBXM7ZM",
    "kind": "definition",
    "msc": [
      "44A15",
      "42A50",
      "30H10"
    ],
    "name": "Hilbert transform",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hilbert_transform"
      }
    ],
    "tag": "HCJK9SV43D",
    "uses": [
      "0SYR10HD4R",
      "050F2FMTKF"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/HCJK9SV43D/",
  "links": [
    {
      "field": "defines",
      "target": "NSCEBXM7ZM"
    },
    {
      "field": "uses",
      "target": "0SYR10HD4R"
    },
    {
      "field": "uses",
      "target": "050F2FMTKF"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/HCJK9SV43D.md",
  "msc": [
    "44A15",
    "42A50",
    "30H10"
  ],
  "name": "Hilbert transform",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/HCJ/HCJK9SV43D.md",
  "status": null,
  "tag": "HCJK9SV43D",
  "uri": "m2://HCJK9SV43D",
  "url": "https://math2.tamnd.com/wiki/hilbert-transform-definition/",
  "writing": {
    "tag": "HCJK9SV43D",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}