{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $(c_n)_{n \\in \\Z}$ be complex numbers, and consider the [[trigonometric series|FZK2M8KGXQ]] $\\sum_{n \\in \\Z} c_n e^{inx}$.\nAssume that for every $x \\in \\R$ its symmetric partial sums converge to $0$:\n$$\\lim_{N \\to \\infty} \\sum_{n=-N}^{N} c_n e^{inx} = 0.$$\nThen $c_n = 0$ for every $n \\in \\Z$.\n\n## Remarks\n\nEquivalently, if two trigonometric series converge at every point to the same sum, then they have the same coefficients.\nSo a function has at most one representation by an everywhere convergent trigonometric series.\n\nProof idea, following Riemann's method.\nFirst, the terms $c_n e^{inx} + c_{-n} e^{-inx}$ tend to $0$ for every $x$, so $c_n \\to 0$ as $\\abs{n} \\to \\infty$ by the [[Cantor–Lebesgue theorem|8Z55VMJSEX]].\nThen the function $F(x) = \\frac{c_0 x^2}{2} - \\sum_{n \\ne 0} \\frac{c_n}{n^2} e^{inx}$ is continuous, and Riemann's first lemma shows that its second symmetric derivative at $x$ equals the sum of the series at $x$, which is $0$.\nA lemma of Schwarz says that a continuous function with vanishing second symmetric derivative is linear, so $F(x) = ax + b$.\nComparing with the periodic part gives $c_0 = 0$, $a = 0$, and then $c_n = 0$ for all $n \\ne 0$ by uniqueness of the coefficients of a uniformly convergent series.\n\nCantor proved the theorem in 1870 and soon allowed finitely many exceptional points.\nIn the language of [[sets of uniqueness|K7CNWXF9SN]], the theorem says that the empty set is a set of uniqueness.\nThe hypothesis of convergence at all points cannot be replaced by convergence almost everywhere.\nMenshov constructed a nonzero trigonometric series that converges to $0$ outside a closed set of measure zero.\n\n## Explanations\n\n### L1 human\n\nIf a sum of waves is equal to $0$ at every point, every single wave in it must be $0$.\nFor finite sums this is easy linear algebra, but for infinite sums that only converge, it is a deep fact.\nIt means that a function can be written as a convergent trigonometric series in at most one way.\n\n### L2 human\n\nThe theorem is about arbitrary trigonometric series, not only Fourier series, so no integral formula for the coefficients is available.\nRiemann's idea of integrating the series twice turns the problem into one about continuous functions.\nTrying to allow larger exceptional sets, Cantor was led to derived sets, transfinite ordinals and set theory.\nThe modern theory of sets of uniqueness continues this question.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "FZK2M8KGXQ",
      "K7CNWXF9SN"
    ],
    "kind": "statement",
    "msc": [
      "42A63",
      "42A20"
    ],
    "name": "Cantor's uniqueness theorem for trigonometric series",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Set_of_uniqueness"
      }
    ],
    "tag": "NSS3XW4BYZ",
    "uses": [
      "FZK2M8KGXQ",
      "8Z55VMJSEX"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/NSS3XW4BYZ/",
  "links": [
    {
      "field": "about",
      "target": "FZK2M8KGXQ"
    },
    {
      "field": "about",
      "target": "K7CNWXF9SN"
    },
    {
      "field": "uses",
      "target": "FZK2M8KGXQ"
    },
    {
      "field": "uses",
      "target": "8Z55VMJSEX"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/NSS3XW4BYZ.md",
  "msc": [
    "42A63",
    "42A20"
  ],
  "name": "Cantor's uniqueness theorem for trigonometric series",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/NSS/NSS3XW4BYZ.md",
  "status": null,
  "tag": "NSS3XW4BYZ",
  "uri": "m2://NSS3XW4BYZ",
  "url": "https://tamnd.github.io/math2/wiki/cantors-uniqueness-theorem-for-trigonometric-series/",
  "writing": {
    "tag": "NSS3XW4BYZ",
    "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
      }
    ]
  }
}