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  "body": "\n## Summary\n\nA [[Fourier series|G12KXDHPPS]] converges absolutely when the sum of the absolute values of its coefficients is finite.\nThen the series converges uniformly to a continuous function, which agrees with the original function almost everywhere.\nIf the function is continuous, it equals the sum of its Fourier series at every point.\nThe functions with this property form the [[Wiener algebra|5H8XESYCHQ]].\nCriteria such as [[Bernstein's theorem|NQC6CRDWFW]] tell which smoothness forces absolute convergence.\n\n## Explanations\n\n### L1 human\n\nA Fourier series writes a periodic function as a sum of waves with amplitudes $c_n$.\nIf the amplitudes are so small that $\\sum \\abs{c_n}$ is finite, the waves can never add up to anything wild.\nThe partial sums then approach the function uniformly, with no Gibbs overshoot.\nFor example, the triangle wave $\\abs{x}$ on $[-\\pi, \\pi]$ has coefficients of size about $1/n^2$, so its Fourier series converges absolutely.\n",
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  "name": "Absolutely convergent Fourier series",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/HHB/HHBGB5R5S2.md",
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