Favard constant

In mathematics, the Favard constant, also called the Akhiezer–Krein–Favard constant, of order r is defined as


 * $$K_r = \frac{4}{\pi} \sum\limits_{k=0}^{\infty} \left[ \frac{(-1)^k}{2k+1} \right]^{r+1}.$$

This constant is named after the French mathematician Jean Favard, and after the Soviet mathematicians Naum Akhiezer and Mark Krein.

Particular values

 * $$K_0 = 1.$$


 * $$K_1 = \frac{\pi}{2}.$$

Uses
This constant is used in solutions of several extremal problems, for example


 * Favard's constant is the sharp constant in Jackson's inequality for trigonometric polynomials
 * the sharp constants in the Landau–Kolmogorov inequality are expressed via Favard's constants
 * Norms of periodic perfect splines.