{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $a \u003c b$ and let $K \\colon [a, b] \\times [a, b] \\to \\C$ be a [[continuous kernel|ZM5G144PZG]] that is [[Hermitian|2ZB3NFZ94T]]: $K(y, x) = \\overline{K(x, y)}$ for all $x, y$.\nLet $(\\varphi_n)_{n \\in N}$ be an orthonormal family of [[eigenfunctions|4NSACSWRJZ]] of $T_K \\varphi = \\int_a^b K(\\cdot, y) \\varphi(y) \\, dy$, with real nonzero eigenvalues $\\mu_n$.\nAssume that, for each nonzero eigenvalue, the $\\varphi_n$ with that eigenvalue form a basis of its eigenspace.\nThese eigenfunctions are continuous, since $\\varphi_n = \\mu_n^{-1} T_K \\varphi_n$.\nLet $h \\in L^2[a, b]$ and $f = T_K h$, that is,\n$$f(x) = \\int_a^b K(x, y) \\, h(y) \\, dy.$$\nThen for every $x \\in [a, b]$,\n$$f(x) = \\sum_{n \\in N} \\ip{f}{\\varphi_n} \\varphi_n(x) = \\sum_{n \\in N} \\mu_n \\ip{h}{\\varphi_n} \\varphi_n(x),$$\nwhere $\\ip{u}{v} = \\int_a^b u \\overline{v}$, and the series converges absolutely and [[uniformly|08VZ57PT26]] on $[a, b]$.\n\n## Remarks\n\nFor fixed $x$, the numbers $\\mu_n \\overline{\\varphi_n(x)}$ are the Fourier coefficients of the function $y \\mapsto \\overline{K(x, y)}$, so [[Bessel's inequality|0RBT5PB74K]] gives $\\sum_n \\mu_n^2 \\abs{\\varphi_n(x)}^2 \\le \\int_a^b \\abs{K(x, y)}^2 \\, dy \\le C$ for a constant $C$ independent of $x$.\nThe Cauchy–Schwarz inequality then bounds the tail $\\sum_{n \\ge m} \\abs{\\mu_n \\ip{h}{\\varphi_n} \\varphi_n(x)}$ by $\\sqrt{C} \\left(\\sum_{n \\ge m} \\abs{\\ip{h}{\\varphi_n}}^2\\right)^{1/2}$, which tends to $0$ uniformly in $x$.\nBy the [[spectral theorem for compact self-adjoint operators|A1G25R99PY]], the series converges to $f$ in $L^2$; the uniform limit is continuous and so is $f$, hence they agree everywhere.\n\nExample: for $K(x, y) = \\min(x, y)$ on $[0, 1]$, the eigenfunctions are $\\sqrt{2} \\sin\\left((n - \\tfrac12) \\pi x\\right)$.\nEvery function $f = T_K h$ with $h \\in L^2[0, 1]$ is continuous with $f(0) = 0$, and its expansion in these sines converges uniformly on $[0, 1]$.\n\nThe hypothesis $f = T_K h$ matters.\nFor a general function $f \\in L^2[a, b]$, the eigenfunction series converges only in $L^2$.\nIts sum is the orthogonal projection of $f$ onto the closed span of the $\\varphi_n$, which may differ from $f$.\n\n## Explanations\n\n### L1 human\n\nThe eigenfunctions of a symmetric kernel are like the axes of an ellipse.\nAny function produced by the integral operator can be written as a sum of these eigenfunctions.\nThe theorem says that this sum converges in the strongest usual sense, uniformly at every point.\n\n### L2 human\n\nThis is the classical form of the [[spectral theorem for compact self-adjoint operators|A1G25R99PY]], adapted to continuous kernels.\nIt is used to solve the Fredholm equation of the second kind with a symmetric kernel: the solution of $\\varphi - \\lambda T_K \\varphi = f$, for $\\lambda$ not a characteristic value, is $\\varphi = f + \\lambda \\sum_n \\frac{\\mu_n \\ip{f}{\\varphi_n}}{1 - \\lambda \\mu_n} \\varphi_n$.\nThrough Green's functions it gives the uniform convergence of eigenfunction expansions for Sturm–Liouville problems.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "4NSACSWRJZ",
      "2ZB3NFZ94T"
    ],
    "kind": "statement",
    "msc": [
      "45C05",
      "47B07"
    ],
    "name": "Hilbert–Schmidt expansion theorem",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hilbert%E2%80%93Schmidt_theorem"
      }
    ],
    "tag": "3CXAZ1G5RJ",
    "uses": [
      "ZM5G144PZG",
      "2ZB3NFZ94T",
      "4NSACSWRJZ",
      "A1G25R99PY",
      "0RBT5PB74K",
      "08VZ57PT26"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/3CXAZ1G5RJ/",
  "links": [
    {
      "field": "about",
      "target": "4NSACSWRJZ"
    },
    {
      "field": "about",
      "target": "2ZB3NFZ94T"
    },
    {
      "field": "uses",
      "target": "ZM5G144PZG"
    },
    {
      "field": "uses",
      "target": "2ZB3NFZ94T"
    },
    {
      "field": "uses",
      "target": "4NSACSWRJZ"
    },
    {
      "field": "uses",
      "target": "A1G25R99PY"
    },
    {
      "field": "uses",
      "target": "0RBT5PB74K"
    },
    {
      "field": "uses",
      "target": "08VZ57PT26"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/3CXAZ1G5RJ.md",
  "msc": [
    "45C05",
    "47B07"
  ],
  "name": "Hilbert–Schmidt expansion theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/3CX/3CXAZ1G5RJ.md",
  "status": null,
  "tag": "3CXAZ1G5RJ",
  "uri": "m2://3CXAZ1G5RJ",
  "url": "https://tamnd.github.io/math2/wiki/hilbert-schmidt-expansion-theorem/",
  "writing": {
    "tag": "3CXAZ1G5RJ",
    "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
      }
    ]
  }
}