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 be a and assume is a positive integer, and that If there exists some fixed number and a sequence of trigonometric polynomials for which and for every
then where the function has a bounded derivative which is -Hölder continuous.