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In [[mathematical analysis]], '''constructive function theory''' is a field which studies the connection between the smoothness of a [[Function (mathematics)|function]] and its degree of [[approximation theory|approximation]].<ref>{{cite web|url=http://encyclopedia2.thefreedictionary.com/Constructive+Theory+of+Functions|title=Constructive Theory of Functions}}</ref><ref>{{SpringerEOM|id=Constructive_theory_of_functions|title=Constructive theory of functions|first=S.A.|last=Telyakovskii}}</ref> It is closely related to [[approximation theory]]. The term was coined by [[Sergei Bernstein]].
 
==Example==
 
Let ''f'' be a 2''π''-periodic function. Then ''f'' is ''α''-[[Hölder condition|Hölder]] for some 0&nbsp;<&nbsp;''α''&nbsp;<&nbsp;1 if and only if for every natural ''n'' there exists a [[trigonometric polynomial]] ''P<sub>n</sub>'' 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)]].
 
==Notes==
{{Reflist}}
 
==References==
 
* {{Cite book |first=N. I. |last=Achiezer |author-link=Naum Akhiezer|title=Theory of approximation |others=Translated by Charles J. Hyman |publisher=Frederick Ungar Publishing |location=New York |year=1956 }}
* {{cite book|mr=0196340|last=Natanson|first=I. P.|author-link=Isidor Natanson|title=Constructive function theory. Vol. I. Uniform approximation|publisher=Frederick Ungar Publishing Co.|location=New York|year=1964}}
: {{cite book|mr=0196341|last=Natanson|first=I. P.|author-link=Isidor Natanson|title=Constructive function theory. Vol. II. Approximation in mean|publisher=Frederick Ungar Publishing Co.|location=New York|year=1965}}
: {{cite book|mr=0196342|last=Natanson|first=I. P.|author-link=Isidor Natanson|title=Constructive function theory. Vol. III. Interpolation and approximation quadratures|publisher=Ungar Publishing Co.|location=New York|year=1965}}
 
[[Category:Approximation theory]]
[[Category:Smooth functions]]

Revision as of 09:03, 9 November 2013

In mathematical analysis, constructive function theory is a field which studies the connection between the smoothness of a function and its degree of approximation.[1][2] It is closely related to approximation theory. The term was coined by Sergei Bernstein.

Example

Let f be a 2π-periodic function. Then f is α-Hölder for some 0 < α < 1 if and only if for every natural n there exists a trigonometric polynomial Pn of degree n such that

max0x2π|f(x)Pn(x)|C(f)nα,

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).

Notes

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