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  "body": "\n## Statement\n\nLet $r \\ge 0$ be an integer.\nFor a continuous $2\\pi$-[[periodic function|0ATF6Q0633]] $g$, let $\\omega(g, \\delta) = \\sup_{\\abs{x - y} \\le \\delta} \\abs{g(x) - g(y)}$ be its [[modulus of continuity|AXCJBSQZWF]].\nFor $n \\ge 1$, let $E_n(f) = \\inf_T \\max_x \\abs{f(x) - T(x)}$, the infimum over real [[trigonometric polynomials|J3NJQ5FZJ5]] $T$ of degree at most $n$.\nThen there is a constant $C_r$, depending only on $r$, such that every $2\\pi$-periodic $f$ with $r$ continuous derivatives satisfies\n$$E_n(f) \\le \\frac{C_r}{n^r} \\, \\omega\\left(f^{(r)}, \\frac{1}{n}\\right) \\qquad \\text{for all } n \\ge 1.$$\n\n## Remarks\n\nProof idea for $r = 0$: approximate $f$ by its convolution with the [[Jackson kernel|9MZFTX77CD]] $J_m$ of degree at most $n$.\nThe kernel is nonnegative with mean $1$, so $\\abs{f - f * J_m} \\le \\frac{1}{2\\pi} \\int \\omega(f, \\abs{t}) J_m(t) \\, dt$.\nWith $\\omega(f, \\abs{t}) \\le (1 + n\\abs{t}) \\, \\omega(f, 1/n)$ and $\\frac{1}{2\\pi} \\int \\abs{t} J_m(t) \\, dt = O(1/n)$, the bound follows.\nThe case $r \\ge 1$ follows by induction on $r$, using $E_n(f) \\le \\frac{C}{n} E_n(f')$ for continuously differentiable $2\\pi$-periodic $f$.\n\nExample: if $f$ is Lipschitz with constant $L$, then $\\omega(f, 1/n) \\le L/n$, so $E_n(f) \\le C_0 L / n$.\n\nThe same bounds hold for algebraic polynomials of degree at most $n$ on $[-1, 1]$ when $r = 0$, through the substitution $x = \\cos \\theta$.\nFor $\\abs{f^{(r)}} \\le 1$ with $r \\ge 1$, the sharp bound is $E_n(f) \\le K_r / (n + 1)^r$, where $K_r$ is the [[Favard constant|9K1W9Q49NR]], by [[Favard's theorem|JYX6N8A2P6]].\n\n## Explanations\n\n### L1 human\n\nThe smoother a periodic function is, the better trigonometric polynomials of a given degree can approximate it.\nJackson's inequality makes this precise: each derivative gains a factor $1/n$, and the modulus of continuity of the last derivative gives the rest.\n\n### L2 human\n\nThis is the basic direct theorem of approximation theory.\nIts converse, Bernstein's inverse theorem, shows that the rates it gives are the right ones when $0 \u003c \\alpha \u003c 1$ in the Hölder scale.\nTogether with [[Bernstein's inequality|F0YGC89S9R]], it underlies the characterisation of smoothness classes by approximation rates.\nFor interpolation and Fourier partial sums, it combines with Lebesgue constants to give convergence results such as the Dini–Lipschitz test.\n",
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