Pickands–Balkema–de Haan 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

$\mathrm{deg}{P}_n=n,{\sup }_{0\le x\le 2\pi }|f(x)-P_n(x)|\le \frac{C(f)}{n^{r+\alpha }},$

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

References

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