Bernstein's theorem (approximation theory)

Template:Short description 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 Template:Nobr be a Template:Nobr and assume Template:Mvar is a positive integer, and that Template:Nobr If there exists some fixed number $k(f)>0$ and a sequence of trigonometric polynomials $(P_{n_0}(x),P_{n_0+1}(x),P_{n_0+2}(x),\dots )$ for which $\mathrm{deg}{P}_n=n$ and $\underset{0\le x\le 2\pi }{sup}|f(x)-P_n(x)|\le \frac{k(f)}{n^{r+\alpha }},$ for every $n\ge n_0,$ then Template:Nobr where the function Template:Math has a bounded Template:Nobr derivative which is [[Hölder condition|Template:Mvar-Hölder continuous]].

See also

• Bernstein's lethargy theorem
• Constructive function theory

References

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