Bernstein's theorem (approximation theory)

{{Short description|In approximation theory, a converse to Jackson's theorem}}

In approximation theory, Bernstein's theorem is a converse to Jackson's theorem.{{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}} The first results of this type were proved by Sergei Bernstein in 1912.{{cite book|last=Bernstein|first=S.N.|author-link=Sergey Bernstein|title=Collected works, 1|year=1952|location=Moscow|pages=11–104}}

For approximation by trigonometric polynomials, the result is as follows:

Let {{nobr| {{math| f: [0, 2π] → ℂ }} }} be a {{nobr|{{math|2 π}} periodic function,}} and assume {{mvar|r}} is a positive integer, and that {{nobr|{{math| 0 < α < 1 }} .}} If there exists some fixed number ~~ k( f ) > 0 ~~ and a sequence of trigonometric polynomials ~~ \Bigl(\ P_{n_0}(x)\ ,\ P_{n_0 + 1}(x)\ ,\ P_{n_0 + 2}(x)\ ,\ \ldots \Bigr) ~~ for which ~~ \deg P_n = n ~~ and ~~ \sup_{0 \leq x \leq 2\pi} \Bigl|f(x) - P_n(x)\Bigr| \leq \frac{\ k(f)\ }{~~ n^{r + \alpha}\ }\ , for every \ n \ge n_0\ ,

then {{nobr| {{math| f(x) {{=}} Pn0(x) + φ(x)}} ,}} where the function {{math|φ(x)}} has a bounded {{nobr|{{mvar|r}} th}} derivative which is Hölder condition.

See also

References

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{{DEFAULTSORT:Bernstein's Theorem (Approximation Theory)}}

Category:Theorems in approximation theory

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