Bernstein's theorem (approximation theory)
{{Short description|In approximation theory, a converse to Jackson's theorem}}
In approximation theory, Bernstein's theorem is a converse to Jackson's theorem.{{cite book|last=Achieser|first=N.I.|author-link=Naum Akhiezer|title=Theory of Approximation|year=1956|publisher=Frederick Ungar Publishing Co|location=New York}} The first results of this type were proved by Sergei Bernstein in 1912.{{cite book|last=Bernstein|first=S.N.|author-link=Sergey Bernstein|title=Collected works, 1|year=1952|location=Moscow|pages=11–104}}
For approximation by trigonometric polynomials, the result is as follows:
Let {{nobr| {{math| f: [0, 2π] → ℂ }} }} be a {{nobr|{{math|2 π}} periodic function,}} and assume {{mvar|r}} is a positive integer, and that {{nobr|{{math| 0 < α < 1 }} .}} If there exists some fixed number and a sequence of trigonometric polynomials for which and for every
then {{nobr| {{math| f(x) {{=}} Pn0(x) + φ(x)}} ,}} where the function {{math|φ(x)}} has a bounded {{nobr|{{mvar|r}} th}} derivative which is Hölder condition.
See also
References
{{reflist|25em}}
{{DEFAULTSORT:Bernstein's Theorem (Approximation Theory)}}
Category:Theorems in approximation theory
{{mathanalysis-stub}}