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  "body": "\n## Summary\n\nAbel–Poisson summation applies [[Abel summation|EZXPY4A19C]] to the [[Fourier series|G12KXDHPPS]] of a periodic function.\nThe $k$-th term is damped by the factor $r^{\\abs{k}}$, and $r$ tends to $1$ from below.\nThe damped series equals the convolution of the function with the [[Poisson kernel|TCYSWCTYYM]].\nAs a function of the point $r e^{ix}$ of the unit disc, the damped sum is a [[harmonic function|0DEB0JMN0W]].\nIts boundary values, in the limit $r \\to 1^-$, are the given function.\nFor a continuous function the Abel–Poisson means converge uniformly to it.\n\n## Explanations\n\n### L1 human\n\nWrite a periodic function as a sum of waves $e^{ikx}$.\nMultiply the wave of frequency $k$ by $r^{\\abs{k}}$ for some $r \u003c 1$, which tames the high frequencies.\nThe result is a smooth function, and it is the steady temperature at radius $r$ in a round plate whose rim is held at the given function.\nAs $r$ goes to $1$, the smoothed function approaches the original one.\n",
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