Bohr-Favard inequality

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

with continuous derivative for given constants and which are natural numbers. The accepted form of the Bohr–Favard inequality is

with the best constant :

The Bohr–Favard inequality is closely connected with the inequality for the best approximations of a function and its -th derivative by trigonometric polynomials of an order at most 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)