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

• Date: 2009
• Main Authors: Kofanov, V. A., Кофанов, В. А.
• Format: Article
• Language: Russian; English
• Published: Institute of Mathematics, NAS of Ukraine 2009
• Online Access: https://umj.imath.kiev.ua/index.php/umj/article/view/3057
• Journal Title: Ukrains’kyi Matematychnyi Zhurnal

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 $L(x)_p ≤ A_0$...