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 in 1912.

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 {Pn}n ≥ n 0 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 = Pn 0 + φ, where φ has a bounded r-th derivative which is α-Hölder continuous.