Favard constant

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

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

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

Particular values

${\displaystyle K_{0}=1.}$

${\displaystyle K_{1}={\frac {\pi }{2}}.}$

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". MathWorld.