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
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with continuous derivative for given constants
and
which are natural numbers. The accepted form of the Bohr–Favard inequality is
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with the best constant :
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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) |
Bohr-Favard inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bohr-Favard_inequality&oldid=17258