Constructive function theory
From HandWiki
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
- [math]\displaystyle{ \max_{0 \leq x \leq 2\pi} | f(x) - P_n(x) | \leq \frac{C(f)}{n^\alpha}, }[/math]
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". http://encyclopedia2.thefreedictionary.com/Constructive+Theory+of+Functions.
- ↑ Hazewinkel, Michiel, ed. (2001), "Constructive theory of functions", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Constructive_theory_of_functions
References
- Achiezer, N. I. (1956). Theory of approximation. New York: Frederick Ungar Publishing.
- Natanson, I. P. (1964). Constructive function theory. Vol. I. Uniform approximation. New York: Frederick Ungar Publishing Co..
- Natanson, I. P. (1965). Constructive function theory. Vol. II. Approximation in mean. New York: Frederick Ungar Publishing Co..
- Natanson, I. P. (1965). Constructive function theory. Vol. III. Interpolation and approximation quadratures. New York: Ungar Publishing Co..
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