{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "R2KZVG9SQ6"
    }
  ],
  "body": "\n## Summary\n\nThese are [[Volterra equations of convolution type|SNFYTC1TTA]] whose kernel $k$ is completely monotone on $(0, \\infty)$: it is infinitely differentiable and its derivatives alternate in sign.\nBy [[Bernstein's theorem on monotone functions|ZBS1JA62QP]], such a kernel is a superposition $k(t) = \\int e^{-st} \\, d\\mu(s)$ of decaying exponentials with a positive measure $\\mu$.\nThe class contains the exponential kernels and the Abel kernels $t^{\\alpha - 1}$ with $0 \u003c \\alpha \u003c 1$.\nIt is a natural setting for positivity and decay results for solutions.\nSuch kernels model fading memory, for instance in linear viscoelasticity.\n\n## Explanations\n\n### L1 human\n\nA completely monotone kernel describes a memory that fades with time in the most regular way possible.\nIt is positive, decreasing, convex, and so on for all higher derivatives.\nEach such kernel is a mixture of simple exponential memories $e^{-st}$ with different decay rates $s$.\nFor example, $k(t) = 1/\\sqrt{t}$ is completely monotone: it is the mixture $\\frac{1}{\\sqrt{\\pi}} \\int_0^\\infty e^{-st} s^{-1/2} \\, ds$.\nSo Abel's integral equation is an equation of this type.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "45D05",
      "45E10",
      "26A48"
    ],
    "name": "Volterra equations with completely monotone kernels",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Bernstein%27s_theorem_on_monotone_functions"
      }
    ],
    "specializes": [
      "SNFYTC1TTA"
    ],
    "tag": "MFJSS5HVA0"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/MFJSS5HVA0/",
  "links": [
    {
      "field": "specializes",
      "target": "SNFYTC1TTA"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/MFJSS5HVA0.md",
  "msc": [
    "45D05",
    "45E10",
    "26A48"
  ],
  "name": "Volterra equations with completely monotone kernels",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/MFJ/MFJSS5HVA0.md",
  "status": null,
  "tag": "MFJSS5HVA0",
  "uri": "m2://MFJSS5HVA0",
  "url": "https://tamnd.github.io/math2/wiki/volterra-equations-with-completely-monotone-kernels/",
  "writing": {
    "tag": "MFJSS5HVA0",
    "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
      }
    ]
  }
}