---
tag: 9K1W9Q49NR
kind: concept
name: Favard constant
msc: [41A44, 42A10, 41A17]
sources:
  - url: https://en.wikipedia.org/wiki/Favard_constant
same_as:
  wikidata: Q5438429
  wikipedia: Favard constant
---

## Summary

The Favard constants $K_0, K_1, K_2, \dots$ are explicit numbers given by alternating series of odd powers.
They are the sharp constants in Jackson's inequality for periodic functions whose $r$-th derivative is bounded by $1$.
They begin with $K_0 = 1$, $K_1 = \pi/2$ and $K_2 = \pi^2/8$, and they tend to $4/\pi$.

## Explanations

### L1 human

If the $r$-th derivative of a periodic function is at most $1$ in size, trigonometric polynomials of degree below $n$ can approximate it within $K_r / n^r$.
The Favard constant $K_r$ is the best number that works here for every such function.
For $r = 1$, a function with slope at most $1$ is within $\pi/(2n)$ of some trigonometric polynomial of degree at most $n - 1$.