Bernstein's theorem (approximation theory)

In approximation theory, Bernstein's theorem is a converse to Jackson's theorem. The first results of this type were proved by Sergei Bernstein.

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 {P_n}_{n ≥ n0} such that

$\deg \,P_{n}=n~,\quad \sup _{0\leq x\leq 2\pi }|f(x)-P_{n}(x)|\leq {\frac {C(f)}{n^{r+\alpha }}}~,$

then f = P_n0 + φ, where φ has a bounded r-th derivative which is α-Hölder continuous.