{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "7HA6EEDR9S"
    },
    {
      "field": "about",
      "target": "CCPS738AH8"
    },
    {
      "field": "uses",
      "target": "7HA6EEDR9S"
    },
    {
      "field": "uses",
      "target": "CCPS738AH8"
    },
    {
      "field": "defines",
      "target": "HZRC5N7M9E"
    }
  ],
  "body": "\n## Summary\n\nA Chebyshev system, or Haar system, is a family of $n$ continuous functions whose nonzero linear combinations have at most $n - 1$ zeros.\nPolynomials of degree less than $n$ are the model case.\nTheir span is called a Haar subspace.\nFor Chebyshev systems, interpolation at any $n$ distinct points is uniquely solvable.\nBy Haar's theorem, they are also the families for which best uniform approximation is always unique.\n\n## Explanations\n\n### L1 human\n\nA polynomial of degree at most $n - 1$ that is not zero crosses the axis at most $n - 1$ times.\nA Chebyshev system is any family of functions that shares this zero-counting property.\nBecause of it, you can always find exactly one combination that passes through $n$ prescribed points with different abscissas.\nExamples are $1, x, \\dots, x^{n-1}$ on an interval, and the functions $1, \\cos x, \\sin x, \\dots, \\cos mx, \\sin mx$ on the circle, with $x$ taken modulo $2\\pi$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "41A50",
      "41A52",
      "41A05"
    ],
    "name": "Chebyshev system",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Equioscillation_theorem"
      }
    ],
    "tag": "NY5AK617TR"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/NY5AK617TR/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/NY5AK617TR.md",
  "msc": [
    "41A50",
    "41A52",
    "41A05"
  ],
  "name": "Chebyshev system",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/NY5/NY5AK617TR.md",
  "status": null,
  "tag": "NY5AK617TR",
  "uri": "m2://NY5AK617TR",
  "url": "https://tamnd.github.io/math2/wiki/chebyshev-system/",
  "writing": {
    "tag": "NY5AK617TR",
    "kind": "concept",
    "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
      }
    ]
  }
}