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  "body": "\n## Summary\n\nThe Dirichlet kernel $D_N(x) = \\sum_{n=-N}^{N} e^{inx}$ is the [[trigonometric polynomial|J3NJQ5FZJ5]] whose [[convolution|2Z29JN3VES]] with a periodic function gives the $N$-th partial sum of its [[Fourier series|G12KXDHPPS]].\nIn closed form $D_N(x) = \\frac{\\sin((N + \\frac{1}{2}) x)}{\\sin(x/2)}$.\nIts integral over a period is constant, but the integral of its absolute value grows like $\\log N$.\nThis growth is the source of most convergence problems of Fourier series, including the existence of continuous functions with divergent Fourier series.\n\n## Explanations\n\n### L1 human\n\nTo get the $N$-th partial Fourier sum of a function at a point, one averages the function against a fixed weight centred at that point, the Dirichlet kernel.\nThe kernel has a tall peak of height $2N + 1$ at $0$ and oscillates with slowly shrinking waves elsewhere, taking negative values.\nBecause of these negative lobes, the averaging is not a true local average, and it can be fooled by functions that oscillate in step with the kernel.\nThe kernel of [[Fejér|DCCA2T9GKH]], an average of Dirichlet kernels, has no negative values and behaves much better.\n",
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