Bohr–Favard inequality
The Bohr–Favard inequality is an inequality appearing in a problem of Harald 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 Jean Favard;[2] the latter materially supplemented the studies of Bohr, and studied the arbitrary periodic function
[math]\displaystyle{ f(x) = \ \sum _ { k=n } ^ \infty (a _ {k} \cos kx + b _ {k} \sin kx) }[/math]
with continuous derivative [math]\displaystyle{ f ^ {(r)} (x) }[/math] for given constants [math]\displaystyle{ r }[/math] and [math]\displaystyle{ n }[/math] which are natural numbers. The accepted form of the Bohr–Favard inequality is
[math]\displaystyle{ \| f \| _ {C} \leq K \| f ^ {(r)} \| _ {C} , }[/math]
[math]\displaystyle{ \| f \| _ {C} = \max _ {x \in [0, 2 \pi ] } | f(x) | , }[/math]
with the best constant [math]\displaystyle{ K = K (n, r) }[/math]:
[math]\displaystyle{ 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]\displaystyle{ r }[/math]th derivative by trigonometric polynomials of an order at most [math]\displaystyle{ n }[/math] and with the notion of Kolmogorov's width in the class of differentiable functions (cf. Width).
References
- ↑ Bohr, Harald (1935). "Un théorème général sur l'intégration d'un polynôme trigonométrique". C. R. Acad. Sci. Paris Sér. I 200: 1276–1277.
- ↑ 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.
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