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 $~~\; \backslash Bigl(\backslash \; P\_\{n\_0\}(x)\backslash \; ,\backslash \; P\_\{n\_0\; +\; 1\}(x)\backslash \; ,\backslash \; P\_\{n\_0\; +\; 2\}(x)\backslash \; ,\backslash \; \backslash ldots\; \backslash Bigr)\; ~~$ for which $~~\; \backslash deg\; P\_n\; =\; n\; ~~$ and $~~\; \backslash sup\_\{0\; \backslash leq\; x\; \backslash leq\; 2\backslash pi\}\; \backslash Bigl|f(x)\; -\; P\_n(x)\backslash Bigr|\; \backslash leq\; \backslash frac\{\backslash \; k(f)\backslash \; \}\{~~\; n^\{r\; +\; \backslash alpha\}\backslash \; \}\backslash \; ,$ for every $\backslash \; n\; \backslash ge\; n\_0\backslash \; ,$

then {{nobr| {{math| f(x) {{=}} P_n0(x) + φ(x)}} ,}} where the function {{math|φ(x)}} has a bounded {{nobr|{{mvar|r}} th}} derivative which is Hölder condition.