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  "body": "\n## Summary\n\nThe Cesàro means of a [[Fourier series|G12KXDHPPS]] are the averages of its first $N$ partial sums.\nThey equal the convolution of the function with the [[Fejér kernel|DCCA2T9GKH]], a positive [[summability kernel|1QC5206K07]].\nFor this reason they behave much better than the partial sums: by [[Fejér's theorem|TPJZCSXSVN]] they converge uniformly for every continuous periodic function.\nThey are the [[Cesàro mean|08R2V9GZF7]] of the sequence of partial sums.\n\n## Explanations\n\n### L1 human\n\nThe partial sums of a Fourier series can oscillate and overshoot, and for some continuous functions they do not converge at all.\nAveraging the first $N$ partial sums smooths out these oscillations.\nIn terms of the waves, the average keeps each frequency $k$ with the weight $1 - \\abs{k}/N$, so high frequencies are faded out gently instead of being cut off.\nFor $f(x) = \\cos x$ the $N$-th mean is $\\frac{N-1}{N} \\cos x$, which tends to $\\cos x$.\n",
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