Английская Википедия:Bohr–Favard inequality

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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

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

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

<math> \| f \| _ {C} \leq K \| f ^ {(r)} \| _ {C} , </math>

<math> \| f \| _ {C} = \max _ {x \in [0, 2 \pi ] } | f(x) | , </math>

with the best constant <math>K = K (n, r)</math>:

<math> K = \sup _ {\| f ^ {(r)} \| _ {C} \leq 1 } \ \| f \| _ {C} . </math>

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

References

Шаблон:Analysis-stub