Bohr–Favard inequality

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

$$ f(x) = \ \sum _ { k=n } ^ \infty (a _ {k} \cos  kx + b _ {k}  \sin  kx) $$

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

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

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

with the best constant $$K = K (n, r)$$:

$$ K =  \sup _ {\| f  ^ {(r)} \| _ {C} \leq  1 } \ \| f \| _ {C}. $$

The Bohr–Favard inequality is closely connected with the inequality for the best approximations of a function and its $$r$$th derivative by trigonometric polynomials of an order at most $$n$$ and with the notion of Kolmogorov's width in the class of differentiable functions (cf. Width).