Constructive function theory: Difference between revisions

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In [[mathematical analysis]], more specifically, in [[approximation theory]], '''constructive function theory''' is a field which studies the connection between the smoothness of a [[function]] and its degree of [[approximation theory|approximation]][http://encyclopedia2.thefreedictionary.com/Constructive+Theory+of+Functions ^{[1<nowiki>]</nowiki>]}[http://eom.springer.de/c/c025430.htm ^{[2<nowiki>]</nowiki>]}. The term was coined by [[Sergei Bernstein]].

==Example==

Let ''f'' be a ''2π''-periodic function. Then ''f'' is ''α''-[[Hölder condition|Hölder]] for some ''0<α<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==

* N.I.Achiezer (Akhiezer), Theory of approximation, Translated by Charles J. Hyman Frederick Ungar Publishing Co., New York 1956 x+307 pp.

[[Category:Approximation theory]]