Constructive function theory: Difference between revisions

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==Example==

Let ''f'' be a 2''2ππ''-periodic function. Then ''f'' is ''α''-[[Hölder condition|Hölder]] for some ''0&nbsp;<&nbsp;''α<1''&nbsp;<&nbsp;1 if and only if for every natural ''n'' there exists a [[trigonometric polynomial]] ''P_n'' of degree ''n'' such that

: <math> \max_{0 \leq x \leq 2\pi} | f(x) - P_n(x) | \leq \frac{C(f)}{n^\alpha}~, </math>

where ''C''(''f)'') is a positive number depending on ''f''. The "only if" is due to [[Dunham Jackson]], see [[Jackson's inequality]]; the "if" part is due to [[Sergei Bernstein]], see [[Bernstein's theorem (approximation theory)]].

==References==