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  "body": "\n## Summary\n\nThe Favard constants $K_0, K_1, K_2, \\dots$ are explicit numbers given by alternating series of odd powers.\nThey are the sharp constants in Jackson's inequality for periodic functions whose $r$-th derivative is bounded by $1$.\nThey begin with $K_0 = 1$, $K_1 = \\pi/2$ and $K_2 = \\pi^2/8$, and they tend to $4/\\pi$.\n\n## Explanations\n\n### L1 human\n\nIf the $r$-th derivative of a periodic function is at most $1$ in size, trigonometric polynomials of degree below $n$ can approximate it within $K_r / n^r$.\nThe Favard constant $K_r$ is the best number that works here for every such function.\nFor $r = 1$, a function with slope at most $1$ is within $\\pi/(2n)$ of some trigonometric polynomial of degree at most $n - 1$.\n",
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