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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 2*π*-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

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).

**^**"Constructive Theory of Functions".**^**Telyakovskii, S.A. (2001) [1994], "Constructive theory of functions",*Encyclopedia of Mathematics*, EMS Press

- 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.