On Kolmogorov-Type Inequalities with Integrable Highest Derivative
Abstract
We obtain the new exact Kolmogorov-type inequality \left\| {x^{\left( k \right)} } \right\|_2 \leqslant K\left\| x \right\|_2^{\frac{{r - k - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}{{r - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}} \left\| {x^{\left( r \right)} } \right\|_1^{\frac{k}{{r{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}}}} for 2π-periodic functions x∈Lr1 and any k, r ∈ N, k < r. We present applications of this inequality to problems of approximation of one class of functions by another class and estimates of K-functional type.Downloads
Published
25.12.2002
Issue
Section
Short communications
How to Cite
Babenko, V. F., et al. “On Kolmogorov-Type Inequalities With Integrable Highest Derivative”. Ukrains’kyi Matematychnyi Zhurnal, vol. 54, no. 12, Dec. 2002, pp. 1694-7, https://umj.imath.kiev.ua/index.php/umj/article/view/4208.