{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "9JVJKJTXDE"
    },
    {
      "field": "illustrates",
      "target": "TB2TR373RC"
    }
  ],
  "body": "\n## Summary\n\nThe Gibbs phenomenon is the overshoot of the partial sums of a [[Fourier series|G12KXDHPPS]] near a [[jump discontinuity|004FH91BN5]] of the function.\nAs more terms are added, the overshoot moves closer to the jump but does not shrink.\nIt tends to about $9\\%$ of the size of the jump on each side.\nIt shows that the [[convergence of a Fourier series|V6Q24CHJG8]] at a jump cannot be uniform.\nIt matters in signal processing, where truncating frequencies causes ringing near sharp edges.\n\n## Explanations\n\n### L1 human\n\nTry to draw a step with a finite number of smooth waves.\nThe flat parts come out nearly right, but just before and after the step the curve shoots past the target and wiggles.\nWith more waves the wiggles get narrower and crowd towards the step, yet the first spike stays about $9\\%$ of the step height too high.\nFor the square wave that jumps from $-1$ to $1$, the partial sums peak near $1.179$ instead of $1$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "42A20",
      "42A16"
    ],
    "name": "Gibbs phenomenon",
    "same_as": {
      "wikidata": "Q5374",
      "wikipedia": "Gibbs phenomenon"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Gibbs_phenomenon"
      }
    ],
    "tag": "VMF169AT67"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/VMF169AT67/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/VMF169AT67.md",
  "msc": [
    "42A20",
    "42A16"
  ],
  "name": "Gibbs phenomenon",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/VMF/VMF169AT67.md",
  "status": null,
  "tag": "VMF169AT67",
  "uri": "m2://VMF169AT67",
  "url": "https://tamnd.github.io/math2/wiki/gibbs-phenomenon/",
  "writing": {
    "tag": "VMF169AT67",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}