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.

References

Weisstein, Eric W. "Favard Constants". MathWorld.

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