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: → C be a 2π-periodic function, and assume r is a natural number, and 0 < α < 1. If there exists a number C > 0 and a sequence of trigonometric polynomials n ≥ n0 such that
then f = Pn0 + φ, where φ has a bounded r-th derivative which is α-Hölder continuous.
