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.

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

${\displaystyle \max _{0\leq x\leq 2\pi }|f(x)-P_{n}(x)|\leq {\frac {C(f)}{n^{\alpha }}},}${\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).

• 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.

• Approximation theory
• Smooth functions