{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Counterexample\n\nLet $\\mathbb{T} = \\R / 2\\pi\\Z$ and let $L^1(\\mathbb{T})$ be the [[space|SASP14MA1Z]] of integrable $2\\pi$-periodic functions.\nFor $f \\in L^1(\\mathbb{T})$ let $S_N f(x) = \\sum_{\\abs{k} \\le N} \\hat f(k) e^{ikx}$ be the partial sums of its [[Fourier series|G12KXDHPPS]], where $\\hat f(k)$ are the [[Fourier coefficients|5NN2YZZR3V]].\nKolmogorov constructed a function $f \\in L^1(\\mathbb{T})$ such that the sequence $S_N f(x)$ diverges for every $x \\in \\R$.\nSo integrability alone gives no [[convergence of the Fourier series|V6Q24CHJG8]] at any point.\n\n## Remarks\n\nKolmogorov first gave, in 1923, an integrable function whose Fourier series diverges almost everywhere, and in 1926 one whose Fourier series diverges everywhere.\nThe construction adds up a series of carefully chosen trigonometric polynomials.\nEach has small $L^1$ norm, but some of its partial sums are very large on most of the circle.\nThe coefficients $\\hat f(k)$ still tend to $0$, by the [[Riemann–Lebesgue lemma|WVKDC011KP]], so the divergence is not caused by large terms.\nThe example shows that [[Carleson's theorem|FBH6KG484Z]] and [[Hunt's extension|NAX51M6025]], which give convergence almost everywhere for $f \\in L^p(\\mathbb{T})$ with $1 \u003c p \u003c \\infty$, have no analogue for $p = 1$.\n\n## Explanations\n\n### L1 human\n\nA Fourier series tries to rebuild a periodic function from pure waves.\nFor square-integrable functions this works at almost every point.\nKolmogorov found an integrable function for which it fails at every single point: the partial sums never settle down anywhere.\nSo the right space for pointwise convergence questions is smaller than the space of all integrable functions.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "counterexample",
    "msc": [
      "42A20",
      "42A16"
    ],
    "name": "Kolmogorov's everywhere divergent Fourier series",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Convergence_of_Fourier_series"
      }
    ],
    "tag": "25V2MHPHZ7",
    "uses": [
      "SASP14MA1Z",
      "G12KXDHPPS",
      "5NN2YZZR3V",
      "V6Q24CHJG8"
    ]
  },
  "kind": "counterexample",
  "link": "https://tamnd.github.io/math2/node/25V2MHPHZ7/",
  "links": [
    {
      "field": "uses",
      "target": "SASP14MA1Z"
    },
    {
      "field": "uses",
      "target": "G12KXDHPPS"
    },
    {
      "field": "uses",
      "target": "5NN2YZZR3V"
    },
    {
      "field": "uses",
      "target": "V6Q24CHJG8"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/25V2MHPHZ7.md",
  "msc": [
    "42A20",
    "42A16"
  ],
  "name": "Kolmogorov's everywhere divergent Fourier series",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/25V/25V2MHPHZ7.md",
  "status": null,
  "tag": "25V2MHPHZ7",
  "uri": "m2://25V2MHPHZ7",
  "url": "https://tamnd.github.io/math2/wiki/kolmogorovs-everywhere-divergent-fourier-series/",
  "writing": {
    "tag": "25V2MHPHZ7",
    "kind": "counterexample",
    "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
      },
      {
        "what": "edge refutes or not_instance_of out",
        "have": 0,
        "want": 1
      }
    ]
  }
}