Favard constant

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

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

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

Particular values

[math]\displaystyle{ K_0 = 1. }[/math]

[math]\displaystyle{ K_1 = \frac{\pi}{2}. }[/math]

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.

References

• Weisstein, Eric W.. "Favard Constants". http://mathworld.wolfram.com/FavardConstants.html.