View source for Bohr–Favard inequality

The '''Bohr–Favard inequality''' is an inequality appearing in a problem of [[Harald Bohr]]<ref name="bohr1935theoreme">{{Cite journal |author=Bohr, Harald |year=1935 |title=Un théorème général sur l'intégration d'un polynôme trigonométrique |journal=C. R. Acad. Sci. Paris Sér. I  |volume=200 |pages=1276–1277}}
</ref> 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 [[Jean Favard]];<ref name="favard1937meilleurs">{{Cite journal |author=Favard, Jean |year=1937 |title=Sur les meilleurs procédés d'approximation de certaines classes des fonctions par des polynômes trigonométriques |journal=Bull. Sci. Math. |volume=61 |pages=243–256 |number=209–224}}
</ref> the latter materially supplemented the studies of Bohr, and studied the arbitrary periodic function

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

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

<math>
\| f \| _ {C}  \leq   K  \| f  ^ {(r)} \| _ {C} ,
</math>

<math>
\| f \| _ {C}  =  \max _ {x \in [0, 2 \pi ] }  | f(x) | ,
</math>

with the best constant <math>K = K (n, r)</math>:

<math>
K  =  \sup _ {\| f  ^ {(r)} \| _ {C} \leq  1 } \ 
\| f \| _ {C} .
</math>

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

==References==
{{reflist}}

{{Free-content attribution
|title= Bohr-Favard inequality
|author = see [https://encyclopediaofmath.org/index.php?title=Bohr-Favard_inequality&action=history revision history] for contributors
|publisher = [[Encyclopedia of Mathematics]]
|documentURL= https://encyclopediaofmath.org/wiki/Bohr-Favard_inequality
|license statement URL= <!-- The URL of the license statement of the work if not included within the document -->
|license= [[CC BY-SA]] and [[GFDL]]
}}

[[Category:Inequalities]]
[[Category:Theorems in real analysis]]


{{Analysis-stub}}