Favard constant

In mathematics, the Favard constant (also called the Akhiezer–Krein–Favard constant) of order ${\displaystyle r}$ is defined as^[1] $${\displaystyle K_{r}={\frac {4}{\pi }}\sum \limits _{k=0}^{\infty }\left[{\frac {(-1)^{k}}{2k+1}}\right]^{r+1}.}$$ The particular values of Favard constant are ${\textstyle K_{0}=1}$, ${\textstyle K_{1}={\frac {\pi }{2}}}$, ${\textstyle K_{2}={\frac {\pi ^{2}}{8}}}$.^[1]

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

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.
• The second Favard constant, ${\displaystyle K_{2}={\frac {\pi ^{2}}{8}}}$  is the same as the value of the internal 4-dimensional equivalent of the angles in a tesseract. The first Favard constant is equal to the value of the internal solid angles in cubes, and the internal angles in squares.^{[citation needed]}

References

1. Finch, Steven R. (2003). Mathematical Constants. Cambridge University Press. p. 256.