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

↑ "Constructive Theory of Functions".
↑ Telyakovskii, S.A. (2001), "Constructive theory of functions", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

References

Achiezer, N. I. (1956). Theory of approximation. Translated by Charles J. Hyman. New York: Frederick Ungar Publishing.
Natanson, I. P. (1964). Constructive function theory. Vol. I. Uniform approximation. New York: Frederick Ungar Publishing Co. MR 0196340.

Natanson, I. P. (1965). Constructive function theory. Vol. II. Approximation in mean. New York: Frederick Ungar Publishing Co. MR 0196341.

Natanson, I. P. (1965). Constructive function theory. Vol. III. Interpolation and approximation quadratures. New York: Ungar Publishing Co. MR 0196342.