Constructive function theory explained

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] 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} | f(x) - P _{n(x)} | \leq (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

• Book: Achiezer, N. I. . Naum Akhiezer. Theory of approximation . Charles J. Hyman . Frederick Ungar Publishing . New York . 1956 .
• Book: Natanson, I. P.. 0196340. Isidor Natanson. Constructive function theory. Vol. I. Uniform approximation. Frederick Ungar Publishing Co.. New York. 1964.

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

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

Notes and References

1. Web site: Constructive Theory of Functions.