2017
Том 69
№ 9

All Issues

On some extremal problems of different metrics for differentiable functions on the axis

Kofanov V. A.

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Abstract

For an arbitrary fixed segment $[α, β] ⊂ R$ and given $r ∈ N, A_r, A_0$, and $p > 0$, we solve the extremal problem $$∫^{β}_{α} \left|x^{(k)}(t)\right|^qdt → \sup,\; q⩾p,\; k=0,\; q⩾1,\; 1 ⩽ k ⩽ r−1,$$ on the set of all functions $x ∈ L^r_{∞}$ such that $∥x (r)∥_{∞} ≤ A_r$ and $L(x)_p ≤ A_0$, where $$L(x)p := \left\{\left( ∫^b_a |x(t)|^p dt\right)^{1/ p} : a,b ∈ R,\; |x(t)| > 0,\; t ∈ (a,b)\right\}$$ In the case where $p = ∞$ and $k ≥ 1$, this problem was solved earlier by Bojanov and Naidenov.

English version (Springer): Ukrainian Mathematical Journal 61 (2009), no. 6, pp 908-922.

Citation Example: Kofanov V. A. On some extremal problems of different metrics for differentiable functions on the axis // Ukr. Mat. Zh. - 2009. - 61, № 6. - pp. 765-776.

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