{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "3QDDSSH4FB"
    },
    {
      "field": "uses",
      "target": "H1HWSWME6B"
    },
    {
      "field": "uses",
      "target": "P3A4XF59FX"
    },
    {
      "field": "uses",
      "target": "T26WGV2PQW"
    },
    {
      "field": "defines",
      "target": "FDF5C62DBD"
    },
    {
      "field": "about",
      "target": "H1HWSWME6B"
    }
  ],
  "body": "\n## Summary\n\nThe iterated kernels of a [[kernel|GSMBXT7FPK]] $K$ are the kernels $K_1 = K$, $K_2$, $K_3$, ... of the powers of its integral operator.\nThey are defined by $K_{n}(x, s) = \\int_a^b K(x, t) K_{n-1}(t, s) \\, dt$, the continuous analogue of matrix powers.\nThey are the terms of the [[Liouville–Neumann series|H1HWSWME6B]] and of the [[resolvent kernel|ARTJV1XCRX]].\nTheir growth in $n$ decides for which $\\lambda$ these series converge.\n\n## Explanations\n\n### L1 human\n\nApplying the integral operator twice is again an integral operator, and its kernel is $K_2$.\nTo compute $K_2(x, s)$, sum over all intermediate points $t$ the product of the influence of $t$ on $x$ and of $s$ on $t$.\nIt works like multiplying a matrix by itself, with the sum over the middle index replaced by an integral.\nFor example, for $K(x, s) = x s$ on $[0, 1]$ one gets $K_2(x, s) = \\int_0^1 x t \\cdot t s \\, dt = x s / 3$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "45B05",
      "45P05"
    ],
    "name": "Iterated kernel",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Liouville%E2%80%93Neumann_series"
      }
    ],
    "specializes": [
      "GSMBXT7FPK"
    ],
    "tag": "9QNBFERMHD"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/9QNBFERMHD/",
  "links": [
    {
      "field": "specializes",
      "target": "GSMBXT7FPK"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/9QNBFERMHD.md",
  "msc": [
    "45B05",
    "45P05"
  ],
  "name": "Iterated kernel",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/9QN/9QNBFERMHD.md",
  "status": null,
  "tag": "9QNBFERMHD",
  "uri": "m2://9QNBFERMHD",
  "url": "https://tamnd.github.io/math2/wiki/iterated-kernel/",
  "writing": {
    "tag": "9QNBFERMHD",
    "kind": "concept",
    "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
      }
    ]
  }
}