# Bohr-Favard inequality

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An inequality appearing in a problem of H. Bohr [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 [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).

#### References

 [1] 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 [2] 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 [3] N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian)
How to Cite This Entry:
Bohr-Favard inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bohr-Favard_inequality&oldid=46096
This article was adapted from an original article by L.V. Taikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article