Bernstein's theorem (approximation theory)

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 2 π periodic function, and assume r is a positive integer, and that 0 < α < 1 . If there exists some fixed number ${\displaystyle ~~k(f)>0~~}${\displaystyle ~~k(f)>0~~} and a sequence of trigonometric polynomials ${\displaystyle ~~{\Bigl (}\ P_{n_{0}}(x)\ ,\ P_{n_{0}+1}(x)\ ,\ P_{n_{0}+2}(x)\ ,\ \ldots {\Bigr )}~~}${\displaystyle ~~{\Bigl (}\ P_{n_{0}}(x)\ ,\ P_{n_{0}+1}(x)\ ,\ P_{n_{0}+2}(x)\ ,\ \ldots {\Bigr )}~~} for which ${\displaystyle ~~\deg P_{n}=n~~}${\displaystyle ~~\deg P_{n}=n~~} and ${\displaystyle ~~\sup _{0\leq x\leq 2\pi }{\Bigl |}f(x)-P_{n}(x){\Bigr |}\leq {\frac {\ k(f)\ }{~~n^{r+\alpha }\ }}\ ,}${\displaystyle ~~\sup _{0\leq x\leq 2\pi }{\Bigl |}f(x)-P_{n}(x){\Bigr |}\leq {\frac {\ k(f)\ }{~~n^{r+\alpha }\ }}\ ,} for every ${\displaystyle \ n\geq n_{0}\ ,}${\displaystyle \ n\geq n_{0}\ ,} then f(x) = P_n0(x) + φ(x) , where the function φ(x) has a bounded r th derivative which is α-Hölder continuous.

• Bernstein's lethargy theorem
• Constructive function theory

• Theorems in approximation theory
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