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  "body": "\n## Statement\n\nLet $r \\ge 1$ and $n \\ge 1$ be integers, and let $K_r$ be the [[Favard constant|9K1W9Q49NR]].\nLet $f \\colon \\R \\to \\R$ be $2\\pi$-[[periodic|0ATF6Q0633]], with $f^{(r-1)}$ absolutely continuous and $\\abs{f^{(r)}} \\le 1$ almost everywhere.\nThen there is a real [[trigonometric polynomial|J3NJQ5FZJ5]] $T$ of degree at most $n - 1$ such that\n$$\\max_{x \\in \\R} \\abs{f(x) - T(x)} \\le \\frac{K_r}{n^r}.$$\nThe constant $K_r$ is sharp.\nFor each $n$, some $f$ of this class has distance exactly $K_r / n^r$ from the set of all such $T$.\n\n## Remarks\n\nThe extremal function is the $r$-fold periodic integral of the square wave $\\sgn \\sin(nx)$, normalised to have mean $0$.\nIts error of best approximation equals $K_r / n^r$, which explains the alternating series in the definition of $K_r$: they come from the Fourier series of the square wave.\nThe upper bound comes from explicit linear approximation methods, so it is attained by a linear operator.\n\nExample: $r = 1$, $n = 1$.\nA periodic function with $\\abs{f'} \\le 1$ has $\\max f - \\min f \\le \\pi$, so the best constant has error at most $\\pi/2 = K_1$.\nThe triangle wave equal to $\\abs{x}$ on $[-\\pi, \\pi]$ reaches this bound.\n\n## Explanations\n\n### L1 human\n\nThis is the best possible form of Jackson's inequality for functions with a bounded $r$-th derivative.\nIt says exactly how well trigonometric polynomials of degree below $n$ can approximate every such function at once.\n\n### L2 human\n\nJackson's inequality gives the rate $n^{-r}$ with some constant; Favard's theorem gives the exact worst-case constant.\nIt is a model result for the theory of widths and optimal recovery, where one asks for the smallest error over a whole class.\nThe same constants appear in sharp inequalities between a function and its derivatives, such as the Kolmogorov inequalities.\n",
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