Favard inequality

The inequality

$$ \tag{* } \| x \| _ {C [ 0, 2 \pi ] } \leq \ M K _ {r} n ^ {-r} ,\ \ r = 1, 2 \dots $$

where

$$ K _ {r} = \ { \frac{4} \pi } \sum _ {k = 0 } ^ \infty (- 1) ^ {k ( r + 1) } ( 2k + 1) ^ {- r - 1 } , $$

and the function $ x ( t) \in W ^ {r} MC $ is orthogonal to every trigonometric polynomial of order not exceeding $ n - 1 $. For $ r = 1 $ inequality (*) was proved by H. Bohr (1935), so it is also called the Bohr inequality and the Bohr–Favard inequality. For an arbitrary positive integer $ r $ inequality (*) was proved by J. Favard [1].

References

[1]	J. Favard, "Sur l'approximation des fonctions périodiques par des polynomes trigonométriques" C.R. Acad. Sci. Paris , 203 (1936) pp. 1122–1124
[2]	V.M. Tikhomirov, "Some problems in approximation theory" , Moscow (1976) (In Russian)

Comments

For a definition of the space $ W ^ {r} MC $ cf. Favard problem.