{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $a \u003c b$.\nA *continuous kernel* on $[a, b]$ is a [[kernel|GSMBXT7FPK]] $K \\colon [a, b] \\times [a, b] \\to \\C$ that is a [[continuous function|0X3WGX95FE]] of the pair $(x, s)$ on the square $[a, b] \\times [a, b]$.\n\n## Remarks\n\nExample: $K(x, s) = \\min(x, s)$ on $[0, 1]$.\nOn the closed triangle $s \\le x$ it equals $s$, and on the closed triangle $x \\le s$ it equals $x$.\nBoth formulas are continuous and agree on the diagonal, so $K$ is continuous on the square.\n\nNon-example: $K(x, s) = \\abs{x - s}^{-1/2}$ for $x \\ne s$, with any value on the diagonal, is not continuous on $[0, 1]^2$, since it is unbounded near the diagonal.\nIt is a weakly singular kernel.\n\nA continuous kernel is bounded, $\\abs{K(x, s)} \\le M$, and uniformly continuous on the compact square.\nSo $x \\mapsto \\int_a^b K(x, s) \\varphi(s) \\, ds$ is continuous for every continuous $\\varphi$, and it is bounded by $M (b - a) \\max \\abs{\\varphi}$.\nThe same definition is used with $[a, b]$ replaced by a compact subset of $\\R^m$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "ZM5G144PZG",
    "kind": "definition",
    "msc": [
      "45P05",
      "45B05"
    ],
    "name": "Continuous kernel",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Integral_equation"
      }
    ],
    "tag": "59CF2D7BRB",
    "uses": [
      "GSMBXT7FPK",
      "0X3WGX95FE"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/59CF2D7BRB/",
  "links": [
    {
      "field": "defines",
      "target": "ZM5G144PZG"
    },
    {
      "field": "uses",
      "target": "GSMBXT7FPK"
    },
    {
      "field": "uses",
      "target": "0X3WGX95FE"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/59CF2D7BRB.md",
  "msc": [
    "45P05",
    "45B05"
  ],
  "name": "Continuous kernel",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/59C/59CF2D7BRB.md",
  "status": null,
  "tag": "59CF2D7BRB",
  "uri": "m2://59CF2D7BRB",
  "url": "https://tamnd.github.io/math2/wiki/continuous-kernel-definition/",
  "writing": {
    "tag": "59CF2D7BRB",
    "kind": "definition",
    "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
      }
    ]
  }
}