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  "body": "\n## Definition\n\nLet $X$ be a normed space of functions and let $V_0 \\subseteq V_1 \\subseteq \\cdots$ be subspaces of $X$.\nFor $f \\in X$ let $E_n(f) = \\inf_{v \\in V_n} \\norm{f - v}$ be the [[error of best approximation|YDN0T4MEX0]].\nLet $\\Omega(f, \\delta)$, for $\\delta \u003e 0$, be a measure of smoothness of $f$, such as a [[modulus of continuity|AXCJBSQZWF]].\n\n1. A *direct theorem* is an inequality $E_n(f) \\le C \\, \\Omega(f, 1/n)$ for all $f \\in X$ and $n \\ge 1$, with $C$ independent of $f$ and $n$.\n2. An *inverse theorem* is an implication that bounds $\\Omega(f, \\delta)$ in terms of the sequence $(E_n(f))_{n \\ge 0}$.\n\n## Remarks\n\nExample: let $X$ be the continuous $2\\pi$-periodic functions with the uniform norm, and $V_n$ the [[trigonometric polynomials|J3NJQ5FZJ5]] of degree at most $n$.\nJackson's inequality $E_n(f) \\le C \\, \\omega(f, 1/n)$, with $\\omega$ the modulus of continuity, is a direct theorem.\nBernstein's theorem, that $E_n(f) \\le M n^{-\\alpha}$ for all $n \\ge 1$ with $0 \u003c \\alpha \u003c 1$ implies $\\omega(f, \\delta) \\le M' \\delta^{\\alpha}$, is an inverse theorem.\nCombined, they say that $E_n(f) = O(n^{-\\alpha})$ if and only if $f$ is [[Hölder continuous|012PBA0FMD]] with exponent $\\alpha$.\n\nNon-example: the bound $E_n(f) \\le \\norm{f}$, from $0 \\in V_n$, is not a direct theorem.\nIt does not involve any smoothness of $f$ and gives no rate as $n \\to \\infty$.\n\nInverse theorems usually rest on inequalities of Bernstein type for the derivatives of elements of $V_n$.\nDirect theorems usually rest on good approximating operators, such as convolution with the Jackson kernel.\n",
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