Pickands–Balkema–de Haan theorem: Difference between revisions
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In [[approximation theory]], '''Bernstein's theorem''' is a converse to [[Jackson's inequality|Jackson's theorem]].<ref>{{cite book|last=Achieser|first=N.I.|author-link=Naum Akhiezer|title=Theory of Approximation|year=1956|publisher=Frederick Ungar Publishing Co|location=New York}}</ref> The first results of this type were proved by [[Sergei Bernstein]] in 1912.<ref>{{cite book|last=Bernstein|first=S.N.|author-link=Sergey Bernstein|title=Collected works, 1|year=1952|location=Moscow|pages=11–104}}</ref> | |||
For approximation by [[trigonometric polynomials]], the result is as follows: | |||
Let ''f'': [0, 2π] → '''C''' be a 2''π''-[[periodic function]], and assume ''r'' is a natural number, and 0 < ''α'' < 1. If there exists a number ''C''(''f'') > 0 and a sequence of [[trigonometric polynomial]]s {''P''<sub>''n''</sub>}<sub>''n'' ≥ ''n''<sub>0</sub></sub> such that | |||
: <math> \deg\, P_n = n~, \quad \sup_{0 \leq x \leq 2\pi} |f(x) - P_n(x)| \leq \frac{C(f)}{n^{r + \alpha}}~,</math> | |||
then ''f'' = ''P''<sub>''n''<sub>0</sub></sub> + ''φ'', where ''φ'' has a bounded ''r''-th derivative which is [[Hölder condition|α-Hölder continuous]]. | |||
==See also== | |||
* [[Bernstein's lethargy theorem]] | |||
* [[Constructive function theory]] | |||
==References== | |||
{{Reflist}} | |||
{{DEFAULTSORT:Bernstein's Theorem (Approximation Theory)}} | |||
[[Category:Approximation theory]] | |||
[[Category:Theorems in approximation theory]] | |||
Revision as of 13:16, 17 September 2013
In approximation theory, Bernstein's theorem is a converse to Jackson's theorem.[1] The first results of this type were proved by Sergei Bernstein in 1912.[2]
For approximation by trigonometric polynomials, the result is as follows:
Let f: [0, 2π] → C be a 2π-periodic function, and assume r is a natural number, and 0 < α < 1. If there exists a number C(f) > 0 and a sequence of trigonometric polynomials {Pn}n ≥ n0 such that
then f = Pn0 + φ, where φ has a bounded r-th derivative which is α-Hölder continuous.
See also
References
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