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).

References

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