{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $V$ be an [[inner product space|0QVNCJXRY3]] with inner product $\\ip{\\cdot}{\\cdot}$, linear in the first argument.\nLet $(\\varphi_k)_{k \\ge 0}$ be a sequence of nonzero, pairwise [[orthogonal|0Y2X7P2WB7]] vectors of $V$.\nFor $f \\in V$, the *generalized Fourier coefficients* of $f$ are the numbers\n$$c_k = \\frac{\\ip{f}{\\varphi_k}}{\\ip{\\varphi_k}{\\varphi_k}}, \\quad k \\ge 0.$$\nThe formal series $\\sum_{k \\ge 0} c_k \\varphi_k$ is the *generalized Fourier series* of $f$ with respect to $(\\varphi_k)$.\n\n## Remarks\n\nIf the $\\varphi_k$ are orthonormal, $c_k = \\ip{f}{\\varphi_k}$.\nThe partial sum $\\sum_{k \\le N} c_k \\varphi_k$ is the orthogonal projection of $f$ onto the span of $\\varphi_0, \\dots, \\varphi_N$, the closest point of that span to $f$.\nBy [[Bessel's inequality|0RBT5PB74K]], $\\sum_k \\abs{c_k}^2 \\norm{\\varphi_k}^2 \\le \\norm{f}^2$.\nIn a [[Hilbert space|HM2HHE3J51]] the series converges to the projection of $f$ onto the closed span of the family.\nIt converges to $f$ for every $f$ exactly when the family is complete, and then [[Parseval's identity|047RFCNVSK]] holds.\n\nExample: in $L^2(-1, 1)$, take the Legendre polynomials $\\varphi_0 = 1$, $\\varphi_1 = x$, $\\varphi_2 = \\frac12 (3x^2 - 1)$, and so on, and $f(x) = x^2$.\nThen $\\ip{f}{\\varphi_0} = \\frac23$, $\\ip{\\varphi_0}{\\varphi_0} = 2$, $\\ip{f}{\\varphi_1} = 0$, $\\ip{f}{\\varphi_2} = \\frac{4}{15}$ and $\\ip{\\varphi_2}{\\varphi_2} = \\frac25$.\nSo $c_0 = \\frac13$, $c_1 = 0$, $c_2 = \\frac23$, and indeed $x^2 = \\frac13 + \\frac23 \\cdot \\frac12(3x^2 - 1)$; all later coefficients vanish.\n\nNon-example: the formula fails to reconstruct $f$ when the family is not orthogonal.\nIn $L^2(0, 1)$ with $\\varphi_0 = 1$ and $\\varphi_1 = x$, which are not orthogonal, the function $f = x$ gives $c_0 = \\frac12$ and $c_1 = 1$, so $c_0 \\varphi_0 + c_1 \\varphi_1 = x + \\frac12 \\ne f$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "SNJC87TBHD",
    "kind": "definition",
    "msc": [
      "42C15",
      "42C10",
      "46C05"
    ],
    "name": "Generalized Fourier series",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Generalized_Fourier_series"
      }
    ],
    "tag": "2S6XZCQWY7",
    "uses": [
      "0QVNCJXRY3",
      "0Y2X7P2WB7",
      "HM2HHE3J51"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/2S6XZCQWY7/",
  "links": [
    {
      "field": "defines",
      "target": "SNJC87TBHD"
    },
    {
      "field": "uses",
      "target": "0QVNCJXRY3"
    },
    {
      "field": "uses",
      "target": "0Y2X7P2WB7"
    },
    {
      "field": "uses",
      "target": "HM2HHE3J51"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/2S6XZCQWY7.md",
  "msc": [
    "42C15",
    "42C10",
    "46C05"
  ],
  "name": "Generalized Fourier series",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/2S6/2S6XZCQWY7.md",
  "status": null,
  "tag": "2S6XZCQWY7",
  "uri": "m2://2S6XZCQWY7",
  "url": "https://tamnd.github.io/math2/wiki/generalized-fourier-series-definition/",
  "writing": {
    "tag": "2S6XZCQWY7",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}