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 Pn of degree n such that
\( {\displaystyle \max _{0\leq x\leq 2\pi }|f(x)-P_{n}(x)|\leq {\frac {C(f)}{n^{\alpha }}},} \)
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) [1994], "Constructive theory of functions", Encyclopedia of Mathematics, EMS Presss
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.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License
