{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $n \\ge 1$ be an integer.\nThe *Chebyshev nodes* of order $n$ in $[-1, 1]$ are the $n$ numbers\n$$x_k = \\cos\\left(\\frac{(2k - 1)\\pi}{2n}\\right), \\qquad k = 1, \\dots, n.$$\nThey are the zeros of the [[Chebyshev polynomial|M6EAR1JHPP]] $T_n$ of the first kind.\nFor an [[interval|0TK8Y3J5RW]] $[a, b]$ with $a \u003c b$, the Chebyshev nodes of order $n$ in $[a, b]$ are the numbers $\\frac{a + b}{2} + \\frac{b - a}{2} x_k$, $k = 1, \\dots, n$.\n\n## Remarks\n\nExample: for $n = 3$ the nodes are $\\cos(\\pi/6), \\cos(\\pi/2), \\cos(5\\pi/6)$, that is $\\sqrt{3}/2$, $0$ and $-\\sqrt{3}/2$.\nCheck: $T_3(x) = 4x^3 - 3x = x(4x^2 - 3)$ vanishes exactly there.\n\nNon-example: the equally spaced points $-1, 0, 1$ are not the Chebyshev nodes of order $3$, since $T_3(1) = 1 \\ne 0$.\n\nThe node polynomial is $\\prod_{k=1}^{n} (x - x_k) = 2^{1-n} T_n(x)$, whose maximum on $[-1, 1]$ is $2^{1-n}$, the least possible for a monic polynomial of degree $n$.\nHence, if $f$ has $n$ continuous derivatives and $p$ interpolates $f$ at these nodes, then $\\max_{[-1, 1]} \\abs{f - p} \\le \\frac{\\max \\abs{f^{(n)}}}{2^{n-1} n!}$.\nThe points $\\cos(k\\pi/n)$, $k = 0, \\dots, n$, where $T_n = \\pm 1$, are the Chebyshev extreme points, also called Chebyshev–Lobatto points.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "HYK8J56PSH",
    "kind": "definition",
    "msc": [
      "65D05",
      "41A05",
      "33C45"
    ],
    "name": "Chebyshev nodes",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Chebyshev_nodes"
      }
    ],
    "tag": "YWTXGDC427",
    "uses": [
      "M6EAR1JHPP",
      "0TK8Y3J5RW"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/YWTXGDC427/",
  "links": [
    {
      "field": "defines",
      "target": "HYK8J56PSH"
    },
    {
      "field": "uses",
      "target": "M6EAR1JHPP"
    },
    {
      "field": "uses",
      "target": "0TK8Y3J5RW"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/YWTXGDC427.md",
  "msc": [
    "65D05",
    "41A05",
    "33C45"
  ],
  "name": "Chebyshev nodes",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/YWT/YWTXGDC427.md",
  "status": null,
  "tag": "YWTXGDC427",
  "uri": "m2://YWTXGDC427",
  "url": "https://tamnd.github.io/math2/wiki/chebyshev-nodes-definition/",
  "writing": {
    "tag": "YWTXGDC427",
    "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
      }
    ]
  }
}