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[edit]

Comments[edit]

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