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

  • V. A. Kofanov

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.
Published
25.06.2009
How to Cite
KofanovV. A. “On Some Extremal Problems of Different Metrics for Differentiable Functions on the Axis”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, no. 6, June 2009, pp. 765-76, http://umj.imath.kiev.ua/index.php/umj/article/view/3057.
Section
Research articles