View source for Bohr-Favard inequality

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An inequality appearing in a problem of H. Bohr [[#References|[1]]] on the boundedness over the entire real axis of the integral of an almost-periodic function. The ultimate form of this inequality was given by J. Favard [[#References|[2]]]; the latter materially supplemented the studies of Bohr, and studied the arbitrary periodic function

$$ 
f(x)  = \ 
\sum _ { k=n } ^  \infty  
(a _ {k}  \cos  kx + b _ {k}  \sin  kx)
$$

with continuous derivative  $  f  ^ {(r)} (x) $
for given constants  $  r $
and  $  n $
which are natural numbers. The accepted form of the Bohr–Favard inequality is

$$ 
\| f \| _ {C}  \leq   K  \| f  ^ {(r)} \| _ {C} ,
$$

$$ 
\| f \| _ {C}  =  \max _ {x \in [0, 2 \pi ] }  | f(x) | ,
$$

with the best constant  $  K = K (n, r) $:

$$ 
K  =  \sup _ {\| f  ^ {(r)} \| _ {C} \leq  1 } \ 
\| f \| _ {C} .
$$

The Bohr–Favard inequality is closely connected with the inequality for the best approximations of a function and its  $  r $-
th derivative by trigonometric polynomials of an order at most  $  n $
and with the notion of Kolmogorov's width in the class of differentiable functions (cf. [[Width|Width]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Bohr,   "Un théorème général sur l'intégration d'un polynôme trigonométrique"  ''C.R. Acad. Sci. Paris Sér. I Math.'' , '''200'''  (1935)  pp. 1276–1277</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Favard,   "Sur les meilleurs procédés d'approximation de certaines classes des fonctions par des polynômes trigonométriques"  ''Bull. Sci. Math. (2)'' , '''61'''  (1937)  pp. 209–224; 243–256</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.I. [N.I. Akhiezer] Achiezer,   "Theory of approximation" , F. Ungar  (1956)  (Translated from Russian)</TD></TR></table>