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  "body": "\n## Definition\n\nLet $f \\colon \\R \\to \\C$ be a $2\\pi$-periodic function that is integrable on $[-\\pi, \\pi]$, with [[Fourier coefficients|5NN2YZZR3V]] $\\hat f(k)$, $k \\in \\Z$.\nFor $n \\ge 0$ let $S_n f(x) = \\sum_{k=-n}^{n} \\hat f(k) e^{ikx}$ be the partial sums of its [[Fourier series|G12KXDHPPS]].\nFor $N \\ge 1$, the $N$-th *Cesàro mean* (or Fejér mean) of the Fourier series of $f$ is\n$$\\sigma_N f(x) = \\frac{1}{N} \\sum_{n=0}^{N-1} S_n f(x).$$\n\n## Remarks\n\nCounting how often each frequency occurs in $S_0 f, \\ldots, S_{N-1} f$ gives\n$$\\sigma_N f(x) = \\sum_{\\abs{k} \u003c N} \\left( 1 - \\frac{\\abs{k}}{N} \\right) \\hat f(k) e^{ikx} = \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} F_N(t) f(x - t) \\, dt,$$\nwhere $F_N$ is the [[Fejér kernel|DCCA2T9GKH]].\nSo $\\sigma_N f$ is a trigonometric polynomial of degree at most $N - 1$.\nIt is the [[Cesàro mean|08R2V9GZF7]] of order $1$ of the sequence $(S_n f(x))_{n \\ge 0}$.\n\nExample: for $f(x) = \\cos x$, $S_0 f = 0$ and $S_n f = \\cos x$ for $n \\ge 1$.\nSo $\\sigma_N f = \\frac{N-1}{N} \\cos x$, which tends to $f$.\n\nNon-example: the Abel–Poisson means $\\sum_k r^{\\abs{k}} \\hat f(k) e^{ikx}$ also average the Fourier series, but with the weights $r^{\\abs{k}}$ instead of $1 - \\abs{k}/N$; they are a different method, the [[Abel–Poisson summation|RZZ78GXP5F]].\n",
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    "name": "Cesàro means of Fourier series",
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