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  "body": "\n## Definition\n\nLet $f$ be a function on $[a, b]$, or a $2\\pi$-periodic function, in a fixed normed space.\nLet $E_n(f)$ be the [[error of best approximation|YDN0T4MEX0]] of $f$ by algebraic, respectively trigonometric, polynomials of degree at most $n$.\n*Constructive function theory* is the part of [[approximation theory|YPJSH79SHQ]] that characterizes classes of functions by the behaviour of the sequence $(E_n(f))_{n \\ge 0}$ as $n \\to \\infty$.\nIts results are of two kinds.\n\n1. Direct theorems: a smoothness condition on $f$, often a bound on its [[modulus of continuity|AXCJBSQZWF]], implies an upper bound for $E_n(f)$.\n2. Inverse theorems: an upper bound for $E_n(f)$ implies a smoothness condition on $f$.\n\n## Remarks\n\nExample: for $2\\pi$-periodic continuous $f$, the uniform norm, and $0 \u003c \\alpha \u003c 1$, one has $E_n(f) = O(n^{-\\alpha})$ if and only if $\\abs{f(x) - f(y)} \\le M \\abs{x - y}^\\alpha$ for some constant $M$ and all $x, y$.\nThe direction from smoothness to the rate is a direct theorem of Jackson type, and the converse is an inverse theorem of Bernstein type.\n\nNon-example: the statement that $E_n(f) \\to 0$ for every continuous $f$ is a result of approximation theory.\nBut it gives no rate and distinguishes no class of functions, so it is not of this kind.\n",
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