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

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

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

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

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

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 ${\displaystyle r}$^th 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).