{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $a \u003c b$ and let $K \\in L^2([a, b] \\times [a, b])$ be a [[square-integrable kernel|WWMWX7AD4V]] that is [[Hermitian|2ZB3NFZ94T]]: $K(y, x) = \\overline{K(x, y)}$ for almost all $x, y$.\nLet $(\\varphi_n)_{n \\in N}$ be an orthonormal family of [[eigenfunctions|4NSACSWRJZ]] of the integral operator $T_K$, with nonzero eigenvalues $\\mu_n$.\nAssume that, for each nonzero eigenvalue, the $\\varphi_n$ with that eigenvalue form a basis of its eigenspace.\nLet $\\lambda_n = 1/\\mu_n$ be the corresponding [[characteristic values|DC7HMHVREP]].\nThen\n$$K(x, y) = \\sum_{n \\in N} \\mu_n \\, \\varphi_n(x) \\overline{\\varphi_n(y)} = \\sum_{n \\in N} \\frac{\\varphi_n(x) \\overline{\\varphi_n(y)}}{\\lambda_n},$$\nwhere the series converges in $L^2([a, b] \\times [a, b])$, and\n$$\\sum_{n \\in N} \\mu_n^2 = \\int_a^b \\int_a^b \\abs{K(x, y)}^2 \\, dx \\, dy.$$\n\n## Remarks\n\nThe functions $\\varphi_n(x) \\overline{\\varphi_m(y)}$ form an orthonormal family in $L^2([a, b]^2)$.\nThe coefficient of $K$ on $\\varphi_n(x) \\overline{\\varphi_m(y)}$ is $\\int \\int K(x, y) \\varphi_m(y) \\overline{\\varphi_n(x)} \\, dy \\, dx = \\mu_m \\ip{\\varphi_m}{\\varphi_n}$, which is $\\mu_n$ if $m = n$ and $0$ otherwise.\nSo the series is the orthogonal projection of $K$ onto the span of this family.\nThe difference between $K$ and the series is a kernel with operator $T_K - \\sum_n \\mu_n \\ip{\\cdot}{\\varphi_n} \\varphi_n$.\nThis operator is $0$ by the [[spectral theorem for compact self-adjoint operators|A1G25R99PY]], so the difference is $0$ almost everywhere.\nThe identity for $\\sum \\mu_n^2$ is then [[Parseval's identity|047RFCNVSK]].\n\nExample: for $K(x, y) = \\min(x, y)$ on $[0, 1]$,\n$$\\min(x, y) = \\sum_{n \\ge 1} \\frac{2 \\sin\\left((n - \\tfrac12)\\pi x\\right) \\sin\\left((n - \\tfrac12)\\pi y\\right)}{(n - \\tfrac12)^2 \\pi^2},$$\nand the identity gives $\\sum_{n \\ge 1} \\frac{1}{(n - \\frac12)^4 \\pi^4} = \\int_0^1 \\int_0^1 \\min(x, y)^2 = \\frac{1}{6}$.\n\nIn general the convergence is only in $L^2$.\nUnder the extra hypotheses of [[Mercer's theorem|PKGADWXPYH]], continuity and positivity of the kernel, the series converges absolutely and uniformly.\n\n## Explanations\n\n### L1 human\n\nA symmetric kernel can be rebuilt from its eigenfunctions and eigenvalues.\nEach eigenfunction contributes one simple product term, weighted by its eigenvalue.\nThis is the integral form of writing a symmetric matrix as a sum of rank-one pieces along its eigenvectors.\n\n### L2 human\n\nThe bilinear series diagonalizes the kernel itself, not only the operator.\nIt gives the estimate $\\sum \\mu_n^2 \u003c \\infty$, so the characteristic values grow fast enough that $\\sum 1/\\lambda_n^2$ converges.\nFor positive continuous kernels it is upgraded to uniform convergence by Mercer's theorem, which also gives the trace formula $\\sum \\mu_n = \\int K(x, x) \\, dx$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "2ZB3NFZ94T"
    ],
    "kind": "statement",
    "msc": [
      "45C05",
      "47B10"
    ],
    "name": "Bilinear series of a symmetric kernel",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Mercer%27s_theorem"
      }
    ],
    "tag": "5WYZPX1ZVW",
    "uses": [
      "WWMWX7AD4V",
      "2ZB3NFZ94T",
      "4NSACSWRJZ",
      "DC7HMHVREP",
      "A1G25R99PY",
      "047RFCNVSK",
      "0S9GGTR0EX"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/5WYZPX1ZVW/",
  "links": [
    {
      "field": "about",
      "target": "2ZB3NFZ94T"
    },
    {
      "field": "uses",
      "target": "WWMWX7AD4V"
    },
    {
      "field": "uses",
      "target": "2ZB3NFZ94T"
    },
    {
      "field": "uses",
      "target": "4NSACSWRJZ"
    },
    {
      "field": "uses",
      "target": "DC7HMHVREP"
    },
    {
      "field": "uses",
      "target": "A1G25R99PY"
    },
    {
      "field": "uses",
      "target": "047RFCNVSK"
    },
    {
      "field": "uses",
      "target": "0S9GGTR0EX"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/5WYZPX1ZVW.md",
  "msc": [
    "45C05",
    "47B10"
  ],
  "name": "Bilinear series of a symmetric kernel",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/5WY/5WYZPX1ZVW.md",
  "status": null,
  "tag": "5WYZPX1ZVW",
  "uri": "m2://5WYZPX1ZVW",
  "url": "https://tamnd.github.io/math2/wiki/bilinear-series-of-a-symmetric-kernel/",
  "writing": {
    "tag": "5WYZPX1ZVW",
    "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
      }
    ]
  }
}