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  "body": "\n## Summary\n\nA summability kernel is a sequence of periodic functions with mean $1$, uniformly bounded integrals of their absolute values, and mass concentrating at $0$.\nConvolving a function with the terms of a summability kernel gives approximations that converge to it in $L^1$.\nFor a continuous function the convergence is uniform.\nThe [[Fejér kernel|DCCA2T9GKH]] and the [[Poisson kernel|TCYSWCTYYM]] are summability kernels, and they explain why [[Cesàro means|MA8X1G11DH]] and Abel means of Fourier series converge.\nThe [[Dirichlet kernel|3JY4BVE4DW]] is not one, which is why ordinary partial sums can fail to converge.\n\n## Explanations\n\n### L1 human\n\nA summability kernel is a family of weights that become a sharper and sharper spike at $0$ while keeping total weight $1$.\nConvolving a function with such a weight replaces each value by an average over a shrinking window.\nSo the averages approach the function itself.\nFor example, the Fejér kernels $F_N$ are nonnegative bumps of mean $1$ whose width near $0$ is about $1/N$.\nThey are the periodic case of an [[approximate identity|CDYXTW3PJF]].\n",
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