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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|Bohr–Favard inequality]]. For an arbitrary positive integer  $  r $
inequality (*) was proved by J. Favard [[#References|[1]]].

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

====Comments====
For a definition of the space  $  W  ^ {r} MC $
cf. [[Favard problem|Favard problem]].