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In approximation theory, **Bernstein's theorem** is a converse to Jackson's theorem.^{[1]} The first results of this type were proved by Sergei Bernstein in 1912.^{[2]}

For approximation by trigonometric polynomials, the result is as follows:

Let *f*: [0, 2π] → **C** be a 2*π*-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

then *f* = *P*_{n0} + *φ*, where *φ* has a bounded *r*-th derivative which is α-Hölder continuous.

**^**Achieser, N.I. (1956).*Theory of Approximation*. New York: Frederick Ungar Publishing Co.**^**Bernstein, S.N. (1952).*Collected works, 1*. Moscow. pp. 11–104.