{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $T_n$ be the [[Chebyshev polynomial|M6EAR1JHPP]] of the first kind of degree $n$.\nLet $p$ be a [[monic|0C3QBCR46M]] real polynomial of [[degree|0DPF6WRZW6]] $n$.\nThen\n$$\\max_{-1 \\le x \\le 1} \\abs{p(x)} \\ge 2^{1-n},$$\nand equality holds if and only if $p = 2^{1-n} T_n$.\n\n## Remarks\n\nProof idea: the polynomial $\\tilde T_n = 2^{1-n} T_n$ is monic and equals $(-1)^k 2^{1-n}$ at the $n + 1$ points $\\cos(k\\pi/n)$, $k = 0, \\dots, n$.\nIf $\\max \\abs{p} \u003c 2^{1-n}$, then $\\tilde T_n - p$ has degree at most $n - 1$ and alternates in sign at these $n + 1$ points.\nSo it has $n$ zeros, hence it is $0$, a contradiction.\nThe equality case follows from a refinement of the same counting, or from the uniqueness of best uniform approximation.\n\nExample: for $n = 2$, $x^2$ has maximum $1$ on $[-1, 1]$, while $\\tilde T_2 = x^2 - 1/2$ has maximum $1/2$, the smallest possible value.\n\nEquivalently, $x^n - \\tilde T_n$ is the best uniform approximation of $x^n$ on $[-1, 1]$ by polynomials of degree at most $n - 1$, with error $2^{1-n}$.\nOn an interval $[a, b]$, an affine change of variable gives the minimum $2 \\left(\\frac{b - a}{4}\\right)^n$.\n\n## Explanations\n\n### L1 human\n\nA monic polynomial has leading term $x^n$, which grows fast, and lower terms can partly cancel it.\nThe theorem says how small such a polynomial can stay on $[-1, 1]$: never below $2^{1-n}$ everywhere.\nThe scaled Chebyshev polynomial is the one that reaches this bound, because it wiggles evenly between $\\pm 2^{1-n}$.\n\n### L2 human\n\nThis is the first and model case of Chebyshev's theory of best uniform approximation.\nIt explains why [[Chebyshev nodes|HYK8J56PSH]] are good for interpolation: the node polynomial $\\prod (x - x_k)$ appears in the interpolation error, and the zeros of $T_n$ make its maximum as small as possible.\nIt also gives the minimax error of $x^n$ and lower bounds for polynomial approximation of smooth functions.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "M6EAR1JHPP"
    ],
    "kind": "statement",
    "msc": [
      "41A50",
      "41A10",
      "33C45"
    ],
    "name": "Minimal deviation property of Chebyshev polynomials",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Chebyshev_polynomials"
      }
    ],
    "tag": "BR3HD0RRA4",
    "uses": [
      "M6EAR1JHPP",
      "0C3QBCR46M",
      "0DPF6WRZW6"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/BR3HD0RRA4/",
  "links": [
    {
      "field": "about",
      "target": "M6EAR1JHPP"
    },
    {
      "field": "uses",
      "target": "M6EAR1JHPP"
    },
    {
      "field": "uses",
      "target": "0C3QBCR46M"
    },
    {
      "field": "uses",
      "target": "0DPF6WRZW6"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/BR3HD0RRA4.md",
  "msc": [
    "41A50",
    "41A10",
    "33C45"
  ],
  "name": "Minimal deviation property of Chebyshev polynomials",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/BR3/BR3HD0RRA4.md",
  "status": null,
  "tag": "BR3HD0RRA4",
  "uri": "m2://BR3HD0RRA4",
  "url": "https://tamnd.github.io/math2/wiki/minimal-deviation-property-of-chebyshev-polynomials/",
  "writing": {
    "tag": "BR3HD0RRA4",
    "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
      }
    ]
  }
}