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
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).
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