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

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

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

$\, f \, _ \leq K \, f ^ \, _ ,$

$\, f \, _ = \max _ , f(x) , ,$

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

$K = \sup _ \ \, f \, _ .$

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

Inequalities
Theorems in real analysis

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