{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a space with a group of symmetries, such as the circle $\\mathbb{T} = \\R / 2\\pi\\Z$ or the space $\\R^n$ with translations.\n*Harmonic analysis* is the branch of analysis that studies functions on such $X$ through the following questions:\n\n1. how a function decomposes into basic oscillations, such as the exponentials $e^{inx}$ on $\\mathbb{T}$ or $e^{2\\pi i x \\cdot \\xi}$ on $\\R^n$;\n2. in which sense, and in which function spaces, the function is recovered from this decomposition;\n3. how operators that commute with translations, such as convolutions and differential operators with constant coefficients, act on the decomposition, and which norm estimates they satisfy.\n\n## Remarks\n\nExample: the theory of [[Fourier series|G12KXDHPPS]] answers these questions on $\\mathbb{T}$, and the [[Fourier transform|DPVM34ADZV]] answers them on $\\R^n$.\nNon-example: the study of the integer solutions of a single polynomial equation by elementary congruences decomposes no function into oscillations, so it is not harmonic analysis.\nThe part over $\\R^n$ and $\\mathbb{T}^n$ is often called Euclidean or classical harmonic analysis.\nThe part over locally compact groups is called abstract harmonic analysis, or [[harmonic analysis on groups|5E6KB3V8R0]].\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "7QNRC9SJVV",
    "kind": "definition",
    "msc": [
      "42A99",
      "42B99",
      "43A99"
    ],
    "name": "Harmonic analysis",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Harmonic_analysis"
      }
    ],
    "tag": "CMMDQ13YEE",
    "uses": [
      "G12KXDHPPS",
      "DPVM34ADZV"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/CMMDQ13YEE/",
  "links": [
    {
      "field": "defines",
      "target": "7QNRC9SJVV"
    },
    {
      "field": "uses",
      "target": "G12KXDHPPS"
    },
    {
      "field": "uses",
      "target": "DPVM34ADZV"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/CMMDQ13YEE.md",
  "msc": [
    "42A99",
    "42B99",
    "43A99"
  ],
  "name": "Harmonic analysis",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/CMM/CMMDQ13YEE.md",
  "status": null,
  "tag": "CMMDQ13YEE",
  "uri": "m2://CMMDQ13YEE",
  "url": "https://tamnd.github.io/math2/wiki/harmonic-analysis-definition/",
  "writing": {
    "tag": "CMMDQ13YEE",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}