---
tag: B9NEZNTRPT
kind: concept
name: Constructive function theory
msc: [41A25, 41A27, 41A10]
sources:
  - url: https://en.wikipedia.org/wiki/Constructive_function_theory
same_as:
  wikidata: Q5164492
  wikipedia: Constructive function theory
---

## Summary

Constructive function theory is the part of [[approximation theory|YPJSH79SHQ]] that describes a function through the speed of its approximation by polynomials.
Its central results show that smoothness of $f$ and fast decay of the [[error of best approximation|YDN0T4MEX0]] $E_n(f)$ go together.
Sergei Bernstein introduced the name for this subject.

## Explanations

### L1 human

A smooth function is easy to imitate with polynomials of low degree, and a rough function needs high degree.
Constructive function theory turns this impression into exact statements.
For example, for a $2\pi$-periodic function and $0 < \alpha < 1$, the best trigonometric approximation of degree $n$ has error of order $n^{-\alpha}$ exactly when the function is Hölder continuous with exponent $\alpha$.
So you can read off the smoothness of a function from how well it can be approximated, and the other way round.