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

$f(x)=\sum _{k=n}^{\mathrm{\infty }}(a_k\cos kx+b_k\sin kx)$

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 $K=K(n,r)$:

$K=\underset{\Vert f^{(r)}{\Vert }_C\le 1}{\sup }\Vert f{\Vert }_C.$

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).

References

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