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  "body": "\n## Counterexample\n\nThere is a [[continuous|0X3WGX95FE]] $2\\pi$-periodic function $f \\colon \\R \\to \\C$ whose [[Fourier series|G12KXDHPPS]] diverges at $0$; in fact $\\sup_N \\abs{S_N f(0)} = \\infty$.\n\nCheck: let $C(\\mathbb{T})$ be the Banach space of continuous $2\\pi$-periodic functions with the supremum norm.\nFor each $N$, the map $\\Lambda_N(f) = S_N f(0) = \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} f(t) D_N(t) \\, dt$ is a bounded linear functional on $C(\\mathbb{T})$, where $D_N$ is the Dirichlet kernel.\nIts norm is the [[Lebesgue constant|RWKQ5EP4YF]] $L_N = \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} \\abs{D_N}$, and $L_N \\to \\infty$.\nIf $\\sup_N \\abs{\\Lambda_N(f)}$ were finite for every $f \\in C(\\mathbb{T})$, the [[uniform boundedness principle|YG0042V8CJ]] would give $\\sup_N L_N \u003c \\infty$, which is false.\nSo some $f \\in C(\\mathbb{T})$ has $\\sup_N \\abs{S_N f(0)} = \\infty$, and its Fourier series diverges at $0$.\n\n## Remarks\n\nThe [[Baire category theorem|0A7KGAYHGV]] behind the uniform boundedness principle gives more: the set of $f \\in C(\\mathbb{T})$ with $\\sup_N \\abs{S_N f(0)} = \\infty$ is a dense $G_\\delta$ subset of $C(\\mathbb{T})$.\nPaul du Bois-Reymond gave the first such function in 1873, by an explicit construction.\nContinuity is thus not enough for pointwise convergence, while bounded variation or a Dini condition is.\nThe divergence can happen only on a set of measure zero for $f \\in L^2$, by [[Carleson's theorem|FBH6KG484Z]].\nBy the [[Kahane–Katznelson theorem|42DDDZHP7F]], every set of measure zero is the divergence set of some continuous function.\nFor integrable functions, [[Kolmogorov's example|25V2MHPHZ7]] diverges everywhere.\n\n## Explanations\n\n### L1 human\n\nOne might expect every continuous periodic function to have a convergent Fourier series, but this is false.\nThe partial sums at a point are averages against a kernel whose total absolute size grows without bound.\nA continuous function that oscillates in step with the kernel, at more and more frequencies, makes these averages grow without bound.\n",
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    "name": "Continuous function with divergent Fourier series",
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  "markdown": "https://tamnd.github.io/math2/node/FPDWWD12FE.md",
  "msc": [
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  "name": "Continuous function with divergent Fourier series",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/FPD/FPDWWD12FE.md",
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