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^[1][2]. It is closely related to approximation theory. The term was coined by Sergei Bernstein.

Example

Let f be a 2π-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 P_n of degree n such that

$\max_{0 \leq x \leq 2\pi} | f(x) - P_n(x) | \leq \frac{C(f)}{n^\alpha},$

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).

Notes

1. ^ "Constructive Theory of Functions". http://encyclopedia2.thefreedictionary.com/Constructive+Theory+of+Functions. 
2. ^ Telyakovskii, S.A. (2001), "Constructive theory of functions", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/c/c025430.htm 

References

• N. I. Achiezer (Akhiezer), Theory of approximation, Translated by Charles J. Hyman Frederick Ungar Publishing Co., New York 1956 x+307 pp.