On Kolmogorov-Type Inequalities with Integrable Highest Derivative

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

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 \(x \in L_1^r\) and any k, rN, 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.
Published
25.12.2002
How to Cite
BabenkoV. F., KofanovV. A., and PichugovS. A. “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.
Section
Short communications