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  "body": "\n## Definition\n\nLet $f \\colon \\R \\to \\C$ be a $2\\pi$-periodic function that is [[integrable|X7GTF3D98F]] on $[-\\pi, \\pi]$.\nFor $n \\in \\Z$ let $\\hat f(n) = \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} f(t) e^{-int} \\, dt$ be its [[Fourier coefficients|5NN2YZZR3V]].\nThe [[Fourier series|G12KXDHPPS]] of $f$ is *absolutely convergent* if\n$$\\sum_{n \\in \\Z} \\abs{\\hat f(n)} \u003c \\infty.$$\n\n## Remarks\n\nIf the series converges absolutely, then $\\sum_n \\hat f(n) e^{inx}$ converges uniformly on $\\R$ to a continuous function $g$.\nThe function $g$ has the same Fourier coefficients as $f$, so $g = f$ almost everywhere.\nOne often says simply that $f$ has an absolutely convergent Fourier series.\n\nExample: let $f(x) = \\abs{x}$ for $-\\pi \\le x \\le \\pi$, extended with period $2\\pi$.\nIntegration by parts gives $\\hat f(0) = \\pi/2$ and $\\hat f(n) = \\frac{(-1)^n - 1}{\\pi n^2}$ for $n \\ne 0$.\nSo $\\abs{\\hat f(n)} \\le 2/(\\pi n^2)$, and the sum $\\sum_n \\abs{\\hat f(n)}$ is finite.\n\nNon-example: the square wave equal to $1$ on $(0, \\pi)$ and $-1$ on $(-\\pi, 0)$ has $\\abs{\\hat f(n)} = \\frac{2}{\\pi \\abs{n}}$ for odd $n$.\nThe sum of these over odd $n$ diverges like the harmonic series.\nThis also follows from the remark above: the square wave is not equal almost everywhere to a continuous function.\n",
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