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]. 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).

References

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