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'': , 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 _{''n'' ≥ ''n''0} such that
: $\deg\, P_n = n~, \quad \sup_ , f(x) - P_n(x), \leq \frac~,$
then ''f'' = ''P''_''n''0 + ''φ'', where ''φ'' has a bounded ''r''-th derivative which is α-Hölder continuous.

See also

* Bernstein's lethargy theorem
* Constructive function theory

References

Theorems in approximation theory

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