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Bernstein's theorem (approximation theory)

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In approximation theory, a converse to Jackson's 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: → 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 {Pn}n ≥ n0 such that

deg P n = n   , sup 0 ≤ x ≤ 2 π | f ( x ) − P n ( x ) | ≤ C ( f ) n r + α   , {\displaystyle \deg \,P_{n}=n~,\quad \sup _{0\leq x\leq 2\pi }|f(x)-P_{n}(x)|\leq {\frac {C(f)}{n^{r+\alpha }}}~,}

then f = Pn0 + φ, where φ has a bounded r-th derivative which is α-Hölder continuous.

See also

References

  1. Achieser, N.I. (1956). Theory of Approximation. New York: Frederick Ungar Publishing Co.
  2. Bernstein, S.N. (1952). Collected works, 1. Moscow. pp. 11–104.{{cite book}}: CS1 maint: location missing publisher (link)


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