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| constructive function theory ： ウィキペディア英語版 |
constructive function theory
In mathematical analysis, constructive function theory is a field which studies the connection between the smoothness of a function and its degree of approximation. It is closely related to approximation theory. The term was coined by Sergei Bernstein.
==Example==
Let ''f'' be a 2''π''-periodic function. Then ''f'' is ''α''-Hölder for some 0 < ''α'' < 1 if and only if for every natural ''n'' there exists a trigonometric polynomial ''Pn'' of degree ''n'' such that
:
where ''C''(''f'') is a positive number depending on ''f''. The "only if" is due to Dunham Jackson, see Jackson's inequality; the "if" part is due to Sergei Bernstein, see Bernstein's theorem (approximation theory).
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