Doob's martingale inequality: Difference between revisions
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In [[mathematics]], the '''Favard constant''', also called the '''Akhiezer–Krein–Favard constant''', of order ''r'' is defined as | |||
:<math>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]]. | |||
==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 spline]]s. | |||
==References== | |||
* {{mathworld|urlname=FavardConstants|title=Favard Constants}} | |||
[[Category:Mathematical constants]] | |||
Revision as of 01:12, 5 May 2013
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}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72fda9996f8d4bdb682099a4b36685088a6163b6)
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.
References
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