Favard constant

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

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

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.

References

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