An extension of the Landau-Kolmogorov inequality. Solution of a problem of Erdös
Tóm tắt
For any fixed finite interval [a, b] on the real line, an arbitrary natural numberr and σ>0, we describe the extremal function to the problem
$$\left\| {f^{(k)} } \right\|L_p \left[ {a,b} \right]^{ \to \sup } \left( {1 \leqslant k \leqslant r - 1, 1 \leqslant p< \infty } \right)$$
over all functionsf ∈W
∞
such that |f
(r)(x)| ≤σ, |f(x)|≤1 on (−∞, ∞). Similarly, we solve the problem, raised by Paul Erdös, of characterizing the trigonometric polynomial of fixed uniform norm whose graph has maximal arc length over [a, b].
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