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 Failed to parse (syntax error): {\displaystyle 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 $\Vert f{\Vert }_C\le K\Vert f^{(r)}{\Vert }_C,$ $\Vert f{\Vert }_C=\underset{x\in [0,2\pi ]}{\max }|f(x)|,$ with the best constant $K=K(n,r)$: Failed to parse (syntax error): {\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 $r$^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).

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