{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $n \\ge 1$, and let $f, g \\colon \\R^n \\to \\C$ be continuous functions with compact supports $\\operatorname{supp} f$ and $\\operatorname{supp} g$.\nHere the support of a function is the closure of the set where it is not $0$.\nThen the [[convolution|2Z29JN3VES]] $f * g$ is continuous, and\n$$\\operatorname{supp}(f * g) \\subseteq \\operatorname{supp} f + \\operatorname{supp} g = \\set{a + b : a \\in \\operatorname{supp} f, \\ b \\in \\operatorname{supp} g}.$$\nIn particular $f * g$ has compact support.\n\n## Remarks\n\nIf $(f * g)(x) = \\int f(y) g(x - y) \\, dy \\ne 0$, then $f(y) g(x - y) \\ne 0$ for some $y$.\nSo $y \\in \\operatorname{supp} f$ and $x - y \\in \\operatorname{supp} g$, and $x = y + (x - y)$ lies in the [[sumset|XNBN31HQWB]] $\\operatorname{supp} f + \\operatorname{supp} g$.\nThis sumset is the image of the [[compact set|0R6EWFKRCF]] $\\operatorname{supp} f \\times \\operatorname{supp} g$ under the continuous map $(a, b) \\mapsto a + b$, so it is compact and hence closed, and it contains the closure of $\\set{f * g \\ne 0}$.\n\nExample: if $f$ and $g$ are continuous, positive on $(0, 1)$ and zero elsewhere, then $f * g$ is positive exactly on $(0, 2)$, and $\\operatorname{supp}(f * g) = [0, 2] = [0, 1] + [0, 1]$.\n\nThe inclusion can be strict, because of cancellation in the integral.\nLet $\\varphi$ be continuous, positive on $(0, 1)$ and zero elsewhere, and put $\\psi = \\varphi * \\varphi$, supported on $[0, 2]$.\nFor $f(x) = \\varphi(x) - \\varphi(x - 2)$ and $g(x) = \\varphi(x) + \\varphi(x - 2)$ the cross terms cancel and $f * g(x) = \\psi(x) - \\psi(x - 4)$.\nSo $\\operatorname{supp}(f * g) = [0, 2] \\cup [4, 6]$, while $\\operatorname{supp} f + \\operatorname{supp} g = [0, 6]$.\nFor integrable functions the same statement holds with essential supports, in the form $\\operatorname{supp}(f * g) \\subseteq \\overline{\\operatorname{supp} f + \\operatorname{supp} g}$.\n\n## Explanations\n\n### L1 human\n\nIf $f$ lives on a set $A$ and $g$ lives on a set $B$, then $f * g$ lives on the set of all sums $a + b$.\nFor instance, two bumps on $[0, 1]$ give a bump on $[0, 2]$.\nSo convolution can spread a function out, but only by a controlled amount.\n\n### L2 human\n\nThis is why convolution with a [[mollifier|CQMK4YPE1J]] supported in a ball of radius $\\varepsilon$ moves the support of a function by at most $\\varepsilon$.\nTogether with [[smoothing by convolution|4P7KPA3Y7R]] it shows that smooth functions with compact support are dense in $L^p$ for $1 \\le p \u003c \\infty$.\nOn the line, the [[Titchmarsh convolution theorem|WGY75H4VWW]] gives an equality for the convex hulls of the supports.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "2Z29JN3VES"
    ],
    "kind": "statement",
    "msc": [
      "44A35",
      "42A85"
    ],
    "name": "Support of a convolution",
    "role": "proposition",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Convolution"
      }
    ],
    "tag": "ZS29A7QQC2",
    "uses": [
      "2Z29JN3VES",
      "0R6EWFKRCF",
      "XNBN31HQWB"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/ZS29A7QQC2/",
  "links": [
    {
      "field": "about",
      "target": "2Z29JN3VES"
    },
    {
      "field": "uses",
      "target": "2Z29JN3VES"
    },
    {
      "field": "uses",
      "target": "0R6EWFKRCF"
    },
    {
      "field": "uses",
      "target": "XNBN31HQWB"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/ZS29A7QQC2.md",
  "msc": [
    "44A35",
    "42A85"
  ],
  "name": "Support of a convolution",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/ZS2/ZS29A7QQC2.md",
  "status": null,
  "tag": "ZS29A7QQC2",
  "uri": "m2://ZS29A7QQC2",
  "url": "https://tamnd.github.io/math2/wiki/support-of-a-convolution/",
  "writing": {
    "tag": "ZS29A7QQC2",
    "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
      }
    ]
  }
}