In approximation theory, Bernstein's theorem is a converse to Jackson's theorem. The first results of this type were proved by Sergei Bernstein in 1912. 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 positive integer, and that 0 < α < 1 . If there exists some fixed number k ( f ) > 0 {\displaystyle ~~k(f)>0~~} and a sequence of trigonometric polynomials ( P n 0 ( x ) , P n 0 + 1 ( x ) , P n 0 + 2 ( x ) , … ) {\displaystyle ~~{\Bigl (}\ P_{n_{0}}(x)\ ,\ P_{n_{0}+1}(x)\ ,\ P_{n_{0}+2}(x)\ ,\ \ldots {\Bigr )}~~} for which deg P n = n {\displaystyle ~~\deg P_{n}=n~~} and sup 0 ≤ x ≤ 2 π | f ( x ) − P n ( x ) | ≤ k ( f ) n r + α , {\displaystyle ~~\sup _{0\leq x\leq 2\pi }{\Bigl |}f(x)-P_{n}(x){\Bigr |}\leq {\frac {\ k(f)\ }{~~n^{r+\alpha }\ }}\ ,} for every n ≥ n 0 , {\displaystyle \ n\geq n_{0}\ ,}
then f(x) = Pn0(x) + φ(x) , where the function φ(x) has a bounded r th derivative which is α-Hölder continuous.
See also Bernstein's lethargy theorem Constructive function theory
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