Exact Kolmogorov-Type Inequalities with Bounded Leading Derivative in the Case of Low Smoothness

  • V. F. Babenko
  • V. A. Kofanov
  • S. A. Pichugov Днепропетр. нац. ун-т ж.-д. трансп.

Abstract

We obtain new unimprovable Kolmogorov-type inequalities for differentiable periodic functions. In particular, we prove that, for r = 2, k = 1 or r = 3, k = 1, 2 and arbitrary q, p ∈ [1, ∞], the following unimprovable inequality holds for functions \(x \in L_\infty ^r \) : $$\left\| {x^{\left( k \right)} } \right\|_q \leqslant \frac{{\left\| {{\phi }_{r - k} } \right\|_q }}{{\left\| {{\phi }_r } \right\|_p^\alpha }}\left\| x \right\|_p^\alpha \left\| {x^{\left( k \right)} } \right\|_\infty ^{1 - \alpha } $$ where \(\alpha = \min \left\{ {1 - \frac{k}{r},\frac{{r - k + {1 \mathord{\left/ {\vphantom {1 q}} \right. \kern-0em} q}}}{{r + {1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-0em} p}}}} \right\}\) and ϕ r is the perfect Euler spline of order r.
Published
25.10.2001
How to Cite
BabenkoV. F., KofanovV. A., and PichugovS. A. “Exact Kolmogorov-Type Inequalities With Bounded Leading Derivative in the Case of Low Smoothness”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 53, no. 10, Oct. 2001, pp. 1299-08, https://umj.imath.kiev.ua/index.php/umj/article/view/4349.
Section
Research articles