{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $T_k$ be the [[Chebyshev polynomials|M6EAR1JHPP]] of the first kind.\nLet $f \\colon [-1, 1] \\to \\R$ be measurable with $\\int_{-1}^{1} \\abs{f(x)} (1 - x^2)^{-1/2} \\, dx \u003c \\infty$.\nThe *Chebyshev coefficients* of $f$ are\n$$a_k = \\frac{2}{\\pi} \\int_{-1}^{1} \\frac{f(x) \\, T_k(x)}{\\sqrt{1 - x^2}} \\, dx, \\qquad k = 0, 1, 2, \\dots$$\nThe *Chebyshev series* of $f$ is the formal series\n$$\\frac{a_0}{2} + \\sum_{k=1}^{\\infty} a_k T_k(x).$$\n\n## Remarks\n\nExample: for $f(x) = x^2$, one finds $a_0 = 1$, $a_2 = 1/2$ and $a_k = 0$ otherwise.\nCheck: $\\frac{2}{\\pi} \\int_{-1}^{1} x^2 (1 - x^2)^{-1/2} \\, dx = \\frac{2}{\\pi} \\cdot \\frac{\\pi}{2} = 1$, and $\\frac{1}{2} + \\frac{1}{2}(2x^2 - 1) = x^2$.\n\nNon-example: $f(x) = (1 - x^2)^{-1/2}$ has no Chebyshev series, since $\\int_{-1}^{1} (1 - x^2)^{-1} \\, dx$ diverges.\n\nWith $x = \\cos \\theta$ and $g(\\theta) = f(\\cos \\theta)$, one gets $a_k = \\frac{2}{\\pi} \\int_0^{\\pi} g(\\theta) \\cos(k\\theta) \\, d\\theta$.\nSo the Chebyshev series of $f$ is the [[Fourier series|G12KXDHPPS]] of the even function $g$, and convergence results transfer.\nFor example, if $f$ is Lipschitz on $[-1, 1]$, the series converges uniformly to $f$.\nThe $T_k$ are [[orthogonal polynomials|8TXASARQN1]] for the weight $(1 - x^2)^{-1/2}$, and, when $\\int_{-1}^{1} f(x)^2 (1 - x^2)^{-1/2} \\, dx \u003c \\infty$, the partial sums are the least squares approximations of $f$ for this weight.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "GP98BJP4JH",
    "kind": "definition",
    "msc": [
      "41A10",
      "42A10",
      "33C45"
    ],
    "name": "Chebyshev series",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Chebyshev_polynomials"
      }
    ],
    "tag": "868V3H5VX8",
    "uses": [
      "M6EAR1JHPP",
      "G12KXDHPPS",
      "8TXASARQN1"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/868V3H5VX8/",
  "links": [
    {
      "field": "defines",
      "target": "GP98BJP4JH"
    },
    {
      "field": "uses",
      "target": "M6EAR1JHPP"
    },
    {
      "field": "uses",
      "target": "G12KXDHPPS"
    },
    {
      "field": "uses",
      "target": "8TXASARQN1"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/868V3H5VX8.md",
  "msc": [
    "41A10",
    "42A10",
    "33C45"
  ],
  "name": "Chebyshev series",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/868/868V3H5VX8.md",
  "status": null,
  "tag": "868V3H5VX8",
  "uri": "m2://868V3H5VX8",
  "url": "https://tamnd.github.io/math2/wiki/chebyshev-series-definition/",
  "writing": {
    "tag": "868V3H5VX8",
    "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
      }
    ]
  }
}