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 (mathematics)|function]] and its degree of [[approximation theory|approximation]][.<ref>{{cite web|url=http://encyclopedia2.thefreedictionary.com/Constructive+Theory+of+Functions|title=Constructive [<sup>1Theory of Functions}}</supref><nowikiref>]{{SpringerEOM|id=Constructive_theory_of_functions|title=Constructive theory of functions|first=S.A.|last=Telyakovskii}}</nowikiref>] It is closely related to [[http://eom.springer.de/c/c025430.htmapproximation [²<nowiki>theory]</nowiki>]. 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_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)]].

==Notes==

{{Reflist}}

==References==

* {{Cite book |first=N. I. |last=Achiezer |author-link=Naum Akhiezer|title=Theory of approximation |translator=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:MathematicalSmooth analysisfunctions]]