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

f ( x ) =   ∑ k = n ∞ ( a k cos ⁡ k x + b k sin ⁡ k x ) {\displaystyle f(x)=\ \sum _{k=n}^{\infty }(a_{k}\cos kx+b_{k}\sin kx)}

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

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

‖ f ‖ C = max x ∈ [ 0 , 2 π ] | f ( x ) | , {\displaystyle \|f\|_{C}=\max _{x\in [0,2\pi ]}|f(x)|,}

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

K = sup ‖ f ( r ) ‖ C ≤ 1   ‖ f ‖ C . {\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 r {\displaystyle r} th derivative by trigonometric polynomials of an order at most n {\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.

 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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