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  "body": "\n## Statement\n\nLet $f \\colon \\R \\to \\C$ be a $2\\pi$-periodic function that is square integrable on $[-\\pi, \\pi]$, that is, $f \\in L^2(\\mathbb{T})$ (see [[Lp space|SASP14MA1Z]]).\nFor $k \\in \\Z$ let $\\hat f(k) = \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} f(t) e^{-ikt} \\, dt$ be its [[Fourier coefficients|5NN2YZZR3V]].\nFor $N \\ge 0$ let $S_N f(x) = \\sum_{k=-N}^{N} \\hat f(k) e^{ikx}$ be the partial sums of its [[Fourier series|G12KXDHPPS]].\nThen $S_N f(x) \\to f(x)$ as $N \\to \\infty$ for [[almost every|RARTMA0ZEC]] $x \\in \\R$.\n\n## Remarks\n\nThe proof reduces the theorem to an estimate for the maximal operator $Cf(x) = \\sup_N \\abs{S_N f(x)}$, namely that it is of weak type $(2, 2)$.\nConvergence holds at every point for trigonometric polynomials, which are dense in $L^2(\\mathbb{T})$, and the maximal estimate passes it to all of $L^2(\\mathbb{T})$.\nCarleson proved the maximal estimate in 1966 by a delicate decomposition of the function in time and frequency.\nLater proofs were given by Fefferman and by Lacey and Thiele.\n\nThe theorem settles a conjecture of Lusin.\nSince continuous periodic functions are square integrable, the Fourier series of a continuous function converges almost everywhere, although it can [[diverge at a point|FPDWWD12FE]].\nThe [[Kahane–Katznelson theorem|42DDDZHP7F]] shows that \"almost everywhere\" cannot be improved, even for continuous functions.\nSquare integrability cannot be weakened to integrability: [[Kolmogorov's example|25V2MHPHZ7]] is an integrable function whose Fourier series diverges everywhere.\n\n## Explanations\n\n### L1 human\n\nThe Fourier series of a periodic function is an infinite sum of waves meant to rebuild the function.\nFor a long time it was unknown whether this sum gives back the function at most points, even for continuous functions.\nCarleson's theorem says yes for every square-integrable function: the sum converges to the right value except on a set of length zero.\n\n### L2 human\n\nConvergence in the $L^2$ norm follows from Parseval's identity and is easy.\nPointwise convergence is much harder because it needs control of all partial sums at once, through the Carleson maximal operator.\n[[Hunt's extension|NAX51M6025]] proves the same for $L^p$ with $1 \u003c p \u003c \\infty$.\nThe time-frequency methods behind the proof became a central tool of modern harmonic analysis.\n",
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