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  "body": "\n## Definition\n\nLet $(c_k)_{k \\in \\Z}$ be complex numbers, and consider the trigonometric series $\\sum_{k \\in \\Z} c_k e^{ikx}$, summed symmetrically as $c_0 + \\sum_{k \\ge 1} (c_k e^{ikx} + c_{-k} e^{-ikx})$.\nFor $0 \\le r \u003c 1$ and real $x$ put\n$$A_r(x) = \\sum_{k \\in \\Z} r^{\\abs{k}} c_k e^{ikx}$$\nwhenever this series converges.\nThe series is *Abel–Poisson summable* at $x$ to $s$ if $A_r(x)$ is defined for all $0 \\le r \u003c 1$ and $A_r(x) \\to s$ as $r \\to 1^-$.\nThis is [[Abel summation|EZXPY4A19C]] of the symmetric series at $x$.\n\n## Remarks\n\nIf $f$ is a [[Riemann integrable|0N1JQXZXHC]] $2\\pi$-[[periodic function|089KWVYD69]] and $c_k = \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} f(t) e^{-ikt} \\, dt$ are its [[Fourier coefficients|G12KXDHPPS]], then the $c_k$ are bounded, $A_r(x)$ converges absolutely, and\n$$A_r(x) = \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} P_r(x - t) f(t) \\, dt, \\qquad P_r(\\theta) = \\sum_{k \\in \\Z} r^{\\abs{k}} e^{ik\\theta} = \\frac{1 - r^2}{1 - 2r\\cos\\theta + r^2}.$$\nHere $P_r$ is the [[Poisson kernel|TCYSWCTYYM]] of the unit disc.\n\nExample: for $f(x) = \\cos x$ one has $c_1 = c_{-1} = \\frac12$ and $c_k = 0$ otherwise, so $A_r(x) = r \\cos x \\to \\cos x$.\n\nNon-example: for $c_k = 1$ for all $k$ and $x = 0$, one gets $A_r(0) = \\sum_{k \\in \\Z} r^{\\abs{k}} = \\frac{1 + r}{1 - r} \\to +\\infty$.\nSo the series $\\sum_{k \\in \\Z} e^{ikx}$ is not Abel–Poisson summable at $0$.\n",
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