{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "MF77N00SK8"
    }
  ],
  "body": "\n## Statement\n\nLet $a \u003c b$.\n\n1. Let $K \\colon [a, b] \\times [a, b] \\to \\C$ be a [[continuous kernel|ZM5G144PZG]], and let $\\varepsilon \u003e 0$. Then there is a [[degenerate kernel|6Y1RY5767R]] $K_\\varepsilon(x, y) = \\sum_{i=1}^{m} a_i(x) b_i(y)$, with polynomials $a_i$ and $b_i$, such that $\\abs{K - K_\\varepsilon} \u003c \\varepsilon$ on $[a, b]^2$.\n2. If $K \\in L^2([a, b] \\times [a, b])$ is a [[square-integrable kernel|WWMWX7AD4V]], then for every $\\varepsilon \u003e 0$ there is a degenerate kernel $K_\\varepsilon$ with $a_i, b_i \\in L^2[a, b]$ such that $\\norm{K - K_\\varepsilon}_{L^2([a, b]^2)} \u003c \\varepsilon$.\n\n## Remarks\n\nPart 1 follows from the [[Stone–Weierstrass theorem|1P8D3MK79H]]: polynomials in $x$ and $y$ are dense in the continuous functions on the square.\nEach monomial $x^p y^q$ is a product of a function of $x$ and a function of $y$.\nFor part 2, let $(e_i)$ be an [[orthonormal basis|0S9GGTR0EX]] of $L^2[a, b]$.\nThe products $e_i(x) \\overline{e_j(y)}$ form an orthonormal basis of $L^2([a, b]^2)$.\nThe partial sums of the expansion of $K$ in this basis are degenerate kernels converging to $K$.\n\nExample: $K(x, y) = e^{xy}$ on $[0, 1]^2$ is approximated by $\\sum_{k=0}^{N} x^k y^k / k!$, with error at most $\\sum_{k \u003e N} 1/k!$.\n\nThe norm of an integral operator is at most the $L^2$ norm of its kernel, and at most $b - a$ times its sup norm.\nSo the result says that the integral operator of $K$ is a norm limit of finite-rank operators.\nThis is the main step in the proof that such operators are compact, and in the reduction of [[Fredholm's theorem|XQ8DYKW4SH]] to linear algebra.\n\n## Explanations\n\n### L1 human\n\nA general kernel mixes the variables $x$ and $y$ in a complicated way.\nA degenerate kernel separates them into finitely many products, and equations with such kernels reduce to finite linear systems.\nThis result says that every reasonable kernel is as close as one wants to a degenerate one.\n\n### L2 human\n\nThe approximation is used together with a perturbation argument.\nWrite $K = K_\\varepsilon + (K - K_\\varepsilon)$; the small part is inverted by a Neumann series, and the degenerate part leads to a finite linear system.\nThis yields the Fredholm alternative for square-integrable kernels, and the compactness of their integral operators.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "6Y1RY5767R"
    ],
    "kind": "statement",
    "msc": [
      "45B05",
      "45P05"
    ],
    "name": "Approximation of a kernel by degenerate kernels",
    "role": "proposition",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Fredholm_integral_equation"
      }
    ],
    "tag": "FJKNJ14AD2",
    "uses": [
      "ZM5G144PZG",
      "WWMWX7AD4V",
      "6Y1RY5767R",
      "1P8D3MK79H",
      "0S9GGTR0EX"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/FJKNJ14AD2/",
  "links": [
    {
      "field": "about",
      "target": "6Y1RY5767R"
    },
    {
      "field": "uses",
      "target": "ZM5G144PZG"
    },
    {
      "field": "uses",
      "target": "WWMWX7AD4V"
    },
    {
      "field": "uses",
      "target": "6Y1RY5767R"
    },
    {
      "field": "uses",
      "target": "1P8D3MK79H"
    },
    {
      "field": "uses",
      "target": "0S9GGTR0EX"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/FJKNJ14AD2.md",
  "msc": [
    "45B05",
    "45P05"
  ],
  "name": "Approximation of a kernel by degenerate kernels",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/FJK/FJKNJ14AD2.md",
  "status": null,
  "tag": "FJKNJ14AD2",
  "uri": "m2://FJKNJ14AD2",
  "url": "https://tamnd.github.io/math2/wiki/approximation-of-a-kernel-by-degenerate-kernels/",
  "writing": {
    "tag": "FJKNJ14AD2",
    "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
      }
    ]
  }
}