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  "body": "\n## Summary\n\nThe Fejér kernel $F_N$ is the average of the first $N$ Dirichlet kernels, the trigonometric polynomials that produce the partial sums of a [[Fourier series|G12KXDHPPS]].\nConvolving a periodic function with $F_N$ gives the $N$-th [[Cesàro mean|08R2V9GZF7]] of its Fourier partial sums.\nUnlike the Dirichlet kernel, the Fejér kernel is nonnegative, has total mass $1$ and concentrates near $0$.\nThese three properties are the heart of [[Fejér's theorem|TPJZCSXSVN]].\n\n## Explanations\n\n### L1 human\n\nA Fourier partial sum is an average of the function against a weight, the Dirichlet kernel, which takes negative values and oscillates strongly.\nAveraging these weights over the first $N$ steps gives the Fejér kernel, which is never negative.\nIts graph is a tall narrow bump at $0$ with small ripples elsewhere, and the bump gets narrower as $N$ grows.\nSo convolving with it computes a local average of the function, which approaches the value of the function at each point of continuity.\n",
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