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. 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 {P_n}_{n ≥ n0} such that

[math]\displaystyle{ \deg\, P_n = n~, \quad \sup_{0 \leq x \leq 2\pi} |f(x) - P_n(x)| \leq \frac{C(f)}{n^{r + \alpha}}~, }[/math]

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

See also

• Bernstein's lethargy theorem
• Constructive function theory