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The Bohr–Favard inequality is an inequality appearing in a problem of Harald Bohr 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; the latter materially supplemented the studies of Bohr, and studied the arbitrary periodic function

${\displaystyle f(x)=\ \sum _{k=n}^{\infty }(a_{k}\cos kx+b_{k}\sin kx)}$

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

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

${\displaystyle \|f\|_{C}=\max _{x\in }|f(x)|,}$

with the best constant ${\displaystyle K=K(n,r)}$:

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

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

References

1. Bohr, Harald (1935). "Un théorème général sur l'intégration d'un polynôme trigonométrique". C. R. Acad. Sci. Paris Sér. I. 200: 1276–1277.
2. Favard, Jean (1937). "Sur les meilleurs procédés d'approximation de certaines classes des fonctions par des polynômes trigonométriques". Bull. Sci. Math. 61 (209–224): 243–256.

[REDACTED]  This article incorporates text from a free content work. Licensed under CC BY-SA and GFDL. Text taken from Bohr-Favard inequality​, see revision history for contributors, Encyclopedia of Mathematics.

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