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  "body": "\n## Summary\n\nConstructive function theory is the part of [[approximation theory|YPJSH79SHQ]] that describes a function through the speed of its approximation by polynomials.\nIts central results show that smoothness of $f$ and fast decay of the [[error of best approximation|YDN0T4MEX0]] $E_n(f)$ go together.\nSergei Bernstein introduced the name for this subject.\n\n## Explanations\n\n### L1 human\n\nA smooth function is easy to imitate with polynomials of low degree, and a rough function needs high degree.\nConstructive function theory turns this impression into exact statements.\nFor example, for a $2\\pi$-periodic function and $0 \u003c \\alpha \u003c 1$, the best trigonometric approximation of degree $n$ has error of order $n^{-\\alpha}$ exactly when the function is Hölder continuous with exponent $\\alpha$.\nSo you can read off the smoothness of a function from how well it can be approximated, and the other way round.\n",
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